Think of space as something entirely smooth that gets wrinkled by objects with mass.
If you roll a ball (or let water flow across it or whatever) along this smooth surface, it will follow the easiest path available, which is along the wrinkles, if that makes sense.
Of course, the image of a plane with dents in it caused by objects with mass is a simplification: that sort of diagram is a 2d world curved into the 3rd dimension. Our universe is 3d and mass curves it into the 4th spatial dimension.
We can't picture 4 dimensions though, which is why we use the simplified image.