There are hundreds of surprising, perspective-shifting insights about the nature of reality that come from neuroscience. Every bizarre neurological syndrome, every visual illusion, and every clever psychological experiment reveals something entirely unexpected about our experience of the world that we take for granted.
Here are a few to give a flavor:

1 Perceptual reality is entirely generated by our brain.
We hear voices and meaning from air pressure waves. We see colors and objects, yet our brain only receives signals about reflected photons. The objects we perceive are a construct of the brain, which is why optical illusions can fool the brain.

2 We see the world in narrow disjoint fragments.
We think we see the whole world, but we are looking through a narrow visual portal onto a small region of space. You have to move your eyes when you read because most of the page is blurry. We don't see this, because as soon as we become curious about part of the world, our eyes move there to fill in the detail before we see it was missing. While our eyes are in motion, we should see a blank blur, but our brain edits this out.

3 Body image is dynamic and flexible.
Our brain can be fooled into thinking a rubber arm or a virtual reality hand is actually a part of our body. In one syndrome, people believe one of their limbs does not belong to them. One man thought a cadaver limb had been sewn onto his body as a practical joke by doctors.

4 Our behavior is mostly automatic, even though we think we are controlling it.
The fact that we can operate a vehicle at 60 mph on the highway while lost in thought shows just how much behavior the brain can take care of on its own. Addiction is possible because so much of what we do is already automatic, including directing our goals and desires. In utilization behavior, people might grab and start using a comb presented to them without having any idea why they are doing it. In impulsivity, people act even though they know they shouldn't.

5 Our brain can fool itself in really strange ways.
In Capgras syndrome, familiar people seem foreign (the opposite of deja vu). One elderly woman who lived alone befriended a woman who appeared to her whenever she looked in a mirror. She thought this other woman looked nothing like herself, except that they seemed to have similar style and tended to wear identical outfits. Another woman was being followed by a tormenter who appeared to her in mirrors but looked nothing like herself. She was fine otherwise.

6 Neurons are really slow.
Our thinking feels fast and we are more intelligent than computers, and yet neurons signal only a few times per second and the brain's beta wave cycles at 14-30 times per second. In comparison, computers cycle at 1 billion operations per second, and transistors switch over 10 billion times per second. How can neurons be so slow and yet we are so smart?

7 Consciousness can be subdivided.
In split-brain patients, each side of the brain is individually conscious but mostly separate from the other. In post-traumatic stress disorder (PTSD) memories of a traumatic event can become a compartmentalized inaccessible island. In schizophrenia, patients hear voices that can seem separate from themselves and which criticize them or issue commands. In hypnosis, post-hypnotic suggestions can direct behavior without the individual's conscious awareness.

That's a glimpse of the world through the eyes of neuroscience.