Most of the explanations of general relativity analogize to a bowling ball's being placed onto a stretched rubber mat. The bowling ball "deforms" the mat, so of course a marble placed onto the mat in the bowling ball's vicinity will roll down the slope in the mat created by the bowling ball towards the bowling ball. No mysterious forces, just geometry.
But when it comes to spacetime itself, what - in lieu of the mat - actually "gets warp?" In the vacuum of space (putting "false vacuum" issues aside) what - in physical terms - is there to be warped in the presence of a massive object. There's no other or other medium, what actually warps?
Great question.
You're looking for something physical, but nothing physically warps. As you said, there's no other or medium.
The "warping" is kind of a metaphor for what happens in a mathematical model. They talk about that and the rubber mat to help people visualize something that's really difficult to grasp, or even believe.
We've all learned from Newton that an object in motion will continue moving in the same direction and speed unless acted upon by another force. Well, you can describe gravity as an outside force, as Newton did, and that's pretty easy to understand. A comet comes flying by the Earth, but the Earth's gravity continually pulls on the comet, and changes its course.
General Relativity takes a different approach to describing the effect of gravity. It says that gravity isn't a force. In its version of reality, the comet keeps moving in a perfectly straight line. Except that a mass can bend the lines. Yeah. No other or medium. But straight lines are bending. It's a model to describe how gravity has the effect it has.
Why do scientists buy into this? Well, experiment has shown that gravity bends light, and our current understanding of light says that light has no mass. So the old model of gravity, which acts only on mass, can't explain this.