How do we know that gravity is spacetime and not a field on spacetime?

Practically speaking, what's the difference? There exists a rank-two tensor field on spacetime called the "metric" $g_{\mu \nu}$ which couples to all mass-energy, and things that we intuitively call "gravity" happen when that field deviates from the Minkowski metric $\eta_{\mu \nu}$. Whether you want to call the metric "spacetime itself" or "a field on spacetime" is basically just terminology.

Steve Weinberg, very unusually, likes to think of general relativity as being just another field theory, with the metric as just another field, and dislikes the "geometric interpretation" that is usually taught, where we describe the metric as being spacetime.

Note, however, that certain stress-energy tensors are only compatible with certain spacetime topologies, so you can't just think of every spacetime as just being $\mathbb{R}^4$ with a funky metric on it.

The reason for regarding gravity specifically as being spacetime, as opposed to any other field, is related to inertia.

Inertia is motion that takes place without forces. So in Newtonian mechanics, inertial motion is moving in a straight line at constant velocity. Mass is the constant of proportionality between the force exerted and the acceleration of the object on which the force acts.

General relativity was motivated in part by Einstein's lift thought experiment. If you're standing in a box being held in a gravitational field, you experience a force opposing the direction of the field as a result of not moving. That is, the floor of the box exerts a force up on your feet when the gravitational field would move you toward the floor. Since gravitational and inertial mass are the same, this is indistinguishable from inertial motion in an accelerating lift. There are some complications like tidal effects, but you can reduce those so they are arbitrarily small by considering a small enough region compared to changes in the gravitational field.

The lift experiment explains that it is physically impossible to distinguish between gravity and inertia. If you attribute inertia to spacetime, then the gravitational field is spacetime, or so the argument goes.