Well, "slows down" is relative terminology, but that says a little more about the terminology than the physics.

The more interesting thing is that "high gravity" is essentially acceleration (and interchangeable with acceleration in most relativity contexts+). This is why the classic "twins on a spaceship" scenario makes the twin on the spaceship the younger one, instead of them being totally interchangeable: the young one is the twin that experiences the acceleration. And you can tell all by yourself that you're undergoing acceleration; you don't need an external reference.

(+ if you accelerate forever, you can even make an acceleration event-horizon. light from beyond this horizon will never actually catch up with you, just like a black hole)

In the "twins on a spaceship" scenario the interchangeability breaks down because there are not two but three relativistic frames - the stationary one; the frame moving away from Earth very fast, and the frame moving towards the Earth very fast. It's not really the acceleration per se, but the combination of these two different frames (of course, switching between them requires changing velocity and thus acceleration) that result in that twin being younger.