The same way(s) you keep any other combat from getting boring.

First, I'm going to point you to DDAL03-04 "Shackles of Blood" which has an excellent arena scene in it exemplifying some of the advice I'll give below. These are presented in ascending order of complexity. Because in your repeated scenario, you should be adding in these elements and combining them in new ways with each iteration.

Terrain

Just because it's an arena doesn't mean that the ground is flat and featureless. Terrain makes non-arena interesting, right? Put some in your arena. This also encompasses hazards and obstructions, not just topography.

Cover & Visibility

Hand-in-hand with terrain, areas of cover and impairments to visibility make tactical decisions more interesting. A literal fog of war changes everything; most importantly, turning combat from a game of perfect-information tactics to a game of imperfect-information risk and resource management. Whether it's weather, terrain, or magic, your ranged attakers should never have the ability to target all points from any one point.

Objectives

Outright slaughter of the opposing side is the least interesting possible setup, yet is too often the go-to. "King of the hill" or "capture the flag" scenarios, a task to be completed while defending an asset, and splintered priorities all spice things up. Is your arena-runner diabolical enough to offer individual rewards to characters, in ways that might put them at odds? Can they make a friend of the slave-captain by comporting themselves in one way, though their safety is dependent on a different behavior?

Time

A ticking clock makes every decision more impactful. Do bridges give way every round, threatening to strand characters far from their allies? Does a new foe enter every N rounds, turning the slower-but-low-risk strategy into a riskier one? Are terrain elements changing with time?

Why focus on combat alone?

The arena is a place where people come to be entertained by gladiatorial displays, chariot racing, brutal executions or whatever else you want to happen there. It is a blank slate to which a D.M. can build an atmosphere of ecstasy, tension within a party, the infamy of a villain, and the downfall of power in one foul swoop. There is plenty here to keep characters and more importantly players evolved for a few levels.

But to address a few quick notes first:

This would also be a good time to introduce your Standard of Play.

Introduce it as the honor rules of the arena. No one speaks while the arena master (you) speak. I am sure there are other clever ways to introduce table etiquette through game play at this stage as well.

You don't need to award experience at the normal rate.

Segment your sessions into lessons you want to teach the new players. Decide how many lessons you want achieved in a session and then award a level for completing the lesson plan.

A few reasons for this:

• Low levels are lethal. We have all been there, made a good character with a decent back story and they die in the first session because they failed a perception check and get back-stabbed by a goblin.
• Low levels are boring. Your characters can't do the cool things yet and can sometimes seem like they are missing more often than not. This can be a drag when it comes to staying engaged with play
• You have a story hook your dying to get your teeth into. This is just the prologue to the story you have been playing. The dramatic set up, you don't want it to go on and on. It'll detract from the story.

Gladiators are expensive.

It costs money to train them, to feed them and to supply them. This is before you take into account the acquisition fees. Have the fights end before anyone dies or has the chance to in post fighting care. Then build to your first kill, it will make it more significant. This is where your description should get gory.

Non-Combat Skills

D&D is more than just a story cobbling together the next big fight. And trying to keep players engaged with combat only can be difficult (unless they are all slayers then throw monster after monster at them)

They will need things to do out of combat, or funky ways to use their skills in combat. A few of these have already been mentioned, like having your rogue unlock a chest of weapons (— keithcurtis, comment section). This is a great example because it is the niche your rogue fills in the party and he is good at it. But you have the potential to use the secondary or ever tertiary skills of the character class. Why not get your fighter to use persuasion/intimidate to roil the crowd before a fight begins. Reward this with a starting round bonus (e.g: advantage on his initiative roll) Try introduce these things before the fight as well, it'll keep new players waiting for the declaration of "roll initiative" to start thinking about what their characters can do.

You can then take skills further away from combat. Get characters use to making lore checks. If the cells are underground, can the dwarf tell you what type of rock it is. Will drinking the water running off the rocks provide you with minerals (giving a small boost to hit dice rolls) or is it to be avoided. What of the furniture, can the ranger tell you the wood it is made from. If so, does he know where this wood comes from. Can that information combined with another characters streetwise create a contact outside?

Remember building up to your first bloody glorious death in the arena? History checks to find out how common this is. Persuasion with the guards to see if they can tell you anything about him. Could a sleight of hand check get the last note he had written to a loved one from the guard while he is distracted. Maybe delivering it in the future with knowledge as to why he died might provide a shelter in a storm later on?

There are many things you could do to introduce story elements for the hook you have planned, or plant the seeds for new hooks. These can all be things to do between the fights (which some really good advice has been given on already). All you need do is look at the skills your characters posses and use the ones that aren't common. It won't matter if it goes nowhere, but the players will enjoy being useful, and who knows it might provide a hook or two.

There's two big things that may help with keeping things fun.

On the Floor

First of all, think of the Colosseum. While when people think of gladiatorial combat, they tend to think 1v1 duel to the death or man vs beast, there would be far more going on than that in an actual arena. Yes, you would have that, but you would also have acrobats and magicians performing routines, chariot races, and stage plays. Foot races, swimming races, and other sporting events. And if your king wants to really pull out all the stops, you can do what Domitian did where he somehow flooded the Colosseum and staged a naval battle.

All those things were done in a world without magic. There's no requirement that your players only compete in the arena. Who's to say a wizard among them wouldn't also be drafted into pleasing the crowd before the individual bouts? The same could be said about rogues and acrobats. Someone got a particularly high CHA score? Maybe they're a comedian or an actor!

Even on the floor itself, you can make things exciting during "traditional" combat. Why not add trees on carts to give cover? Why not allow the whole party to sometimes compete together and face off another band of criminals so they can learn the value of teamwork in D&D?

If you're cool with homebrew, you might take a look at this idea of turning Witch Bolt into a dueling spell. It might not be right for your world, but I guarantee watching two wizards go at it with a genuine dueling spell would have the arena's crowd spellbound (pun very much intended).

Variety is the spice of life, so as long as you go big (and since they'll eventually be granted a chance at freedom by the king himself, it sounds like you can), you shouldn't have any difficulty keeping what's happening on the arena floor itself interesting and fun.

Off the Floor

The other thing is that what goes on on the floor is only a fraction of the whole gladiator lifestyle. Clearly, it's the exciting part, but it's not the only part. Suppose immediately following a bout, the victor(s) get to make a victory lap during which the crowd is free to donate coins to the cause. Now your gladiators have a source of income they can use to buy better gear, bribe guards, and secure niceties they might not otherwise have.

Now that they've got an income, what if they were allowed to place bets on the victors in NPC fights? What if they could surreptitiously interfere with future opponents, fiddling with their gear to make things go in their favor?

What about the social dynamics of getting in line for the daily slop? Those perceived as the best or strongest and those they like might get the front of the line while others find their way toward the back where they'll get less food and it'll be colder. Does your party band together to try to collectively bargain for a higher rank in the line? Does the tiefling barbarian muscle his way through and the gnome he's friends with follows in his wake? Does the silver-tongued halfling rogue just talk his way to the front of the line? Does the particularly charismatic fighter who receives an unfairly large sum of money from the crowd just bribe her way to the front?

What about practice? Gladiators didn't just randomly get pulled out of cells and tossed into the arena (obviously some did, but not all, and certainly not those that made a career of it, which yours effectively are, it sounds like). You could allow them to train against each other with nonlethal weapons, still giving them experience without putting their lives in danger. This would also give them a chance to show off their skills to the other gladiators. Maybe a few NPCs will become sycophants to your party. Maybe one might even get released alongside your party and join them or others might show up later in their quest after securing their own releases.

Maybe there's already a dominant gladiator in their cell block with a following. Do the PCs beat him into submissions? Kill him quietly in the night? Bribe him? Join him? Convince him to join them? He could even effectively be the Big Bad for this part of the story if that's something you'd like, a way to give it an overall arc without really impeding the rest of the story or preventing it from progressing forward. Maybe beating him and his goons in the arena is the final battle that gets the king's favor?

Conclusion

Whatever you do, just make sure to keep things variable and make sure the players are doing more than just fighting increasing numbers of goblins. There's a ton you can do on and off the floor to keep their lives interesting, letting them learn about the game at their own pace, and giving them clear goals to work toward.

Another thing you may want to consider:

The Crowd

The audience is a huge deal for arena combat. Sometimes, the crowd might even rush the field. They could throw things. They can mob and assault after the match and destroy things in town.

Read about the Greens and the Blues (and the Reds and the Whites) who were team fans of the chariot races in Constantinople. They altered politics with their mob actions.

Speaking of? Mix up events. Have naval battles (flooded arena), chariot races, historical battles, etc.

Matthew Mercer runs a very entertaining arena combat in episode 17 of Critical Role.

The key tools he uses are:

Very visceral description of the damage being done. This might require some off-the-cuff usage of hit locations and penalties that are not part of the ordinary rules. This will heighten the excitement by focusing on the combat itself instead of the rote mechanics of dice rolling. Be free with inspiration to encourage creative description from the players as well

Allowing non-combatants to be involved in a tangential manner. They can shout advice to give inspiration or advantage as abilities allow. They can place bets, or trash talk the opponent. This will keep non-fighting players from getting sidelined and bored.

Use no magic. This makes it more of a brute force contest. In his arena combat, Mercer uses no weapons, either, with damage being done by open fist plus STR mod. "No magic" might be going in a different direction than you intend though.

I would highly recommend listening to this episode to see how arena combat can be made entertaining.