It means this:

in newtonian mechanics, a free body in an inertial ref. frame (free = no forces on it) moves with constant velocity. Let's imagine, e.g., a coin forced to move, without friction, on a flat table. It will keep moving forever, undisturbed. In this sense we can say that it follows the geometry of space, that is, in this case, the plane of the table.

Now imagine that the table is not flat anylonger, but is a spherical surface covering our planet earth. Observing locally the body's behaviour, you will see that it still moves with constant velocity, if you don't apply forces on it.

You could say: "one moment!" "the velocity is not constant anylonger now, because the velocity changes in direction, along the spherical surface"

But, to say that, you have to move away from the surface, and see things from the 3rd dimension.

If we were beings living in only the 2 dimensional spherical surface, we couldn't say that.

Now, what happens is that we live in a 3 dimensional space, but this is not flat, is warped! To see it, we should be able to see things from the 4th dimension, but, unfortunately, we can't!

The fact is that according to Einstein's GR, gravity (produced by: mass, energy, momentum) warps the 4-dimensional space-time so that we are forced to follow that " 4-D analogus of a 2-D warped surface". So, from that 4-D point of view, we are not experiencing any acceleration, in absence of other forces (for example, electrical), but we simply "keep moving freely" on that warped space.