You're right that you can think of gravity as something that isn't a force, but it works differently than what you describe in your post.

Essentially, an object with no (non-gravity) forces on it will trace out a straight line through space-time. This is basically inertia: an object won't change speed unless some force acts on it, and if you chart out the path an object moving at a constant speed traces, it follows a straight line. Technically, it's called a "geodesic," not a straight line, but that's essentially the general concept that includes "straight lines."

Gravity happen because geodesics tend to get closer to massive objects. This seems strange; for instance, if you were on a spaceship moving next to the earth at exactly the same speed in exactly the same direction, the paths you and the earth are taking would be parallel. There'd be no significant forces affecting your ship and the earth, so you'd expect those forces to stay parallel since you're moving in a straight line, but your spaceship would get closer and closer to the earth until you eventually crash. This is because spacetime isn't "flat," and geometry in curved spaces behaves differently than geometry in flat ones. As an example, if you take a globe and draw two lines next to each other, perpendicular to the equator, those lines look parallel. But if you continue to draw the lines straight (technically, drawing geodesics on a sphere), they would intersect at the north and south poles. That's because the surface of a globe is curved in a way that "parallel" lines eventually cross.

Something similar happens with spacetime. Mass (and energy) itself causes spacetime to curve. It doesn't seem to curve into anything, in the same way that a globe curves in 3d space, it just sort of intrinsically is curved. We also don't really know why energy curves space, beyond "it just does." But it does curve spacetime, and it curves it in such a way that the geodesics that other objects follow tend to get closer to it over time. Heavier things curve spacetime more, which makes geodesics get closer in a more pronounced way, which makes objects fall faster.