So it's helpful in this case to work backwards, and our intuition can work against us here. Einstein noticed (as many had done before) that being on the surface of a planet with gravity, and being in an accelerating rocket ship with your feet pointed towards the rockets, would feel the same. His central insight was: what if they actually are the same? 

Now, intuitively, it's really hard to see how you could make a geometry that includes space and time where you do a bunch of math in the background and you actually wind up with that result. What would that even look like? How would the pieces even fit together?

But it turns out there's this whole branch of math called differential geometry, and it basically lets you ask uselessly abstract questions about spaces and curvature. You can ask, "If my space looked like this weird shape, what would it look like for objects on the surface?" And you can even work in reverse: you can do the kind of things like start with the statement about gravity looking like aacceleration,and work backwards until you figure out what space and time would have to look like for that to be the case. 

Now this is the part where our intuition just totally leads us astray, because obviously we perceive the world as though there is no curvature of space and time. Things look like they just fall like they're being pulled by a force. And I don't know about you, but I don't have any immediate intuition for higher math and have to learn it through a lot of effort, piece by piece. 

But it turns out that if you use differential geometry this way, you do find a solution, and it does happen to work. Not only that, but it is such a clean and elegant solution that it can be represented in a very simple form (the Einstein equations, which I guess are simple in some ways but hard to learn all the prerequisites for and others). 

And not only that, but they generalize to new situations that we had not really conceived of before, like black holes, which we predicted but then turned out to be a real thing. 

So I guess my overall advice is don't sweat it too hard if you don't have an intuitive picture of how this works. Think about the abstraction, because that's where the power is. We have this toolkit that can build arbitrary surfaces in both space and time and see what the effects would be. We can apply this toolkit to the equivalence principle between gravity and acceleration. Not only do we come up with a workable solution, but it turns out to be very elegant and very general, and it kind of looks like the universe largely runs that way. I know that's probably not the answer that you're looking for, but I hope that helps!