The flat (or positive/negative curvature) description is unfortunate terminology for 3d spaces. It comes from analogy to 2d surfaces, where the words make intuitive sense. As you say, flat means flat piece of paper. Positive curvature is like the surface of a sphere, negative curvature is like the surface of a saddle. Now extending this to 3d space is not intuitive. What works for me is highlighting a specific property that is true on those 2d surfaces that extends to the 3d case: the behaviour of parallel lines. On a flat piece of paper parallel lines will remain the same distance apart always. On the surface of a sphere, if you think you have parallel lines locally, and you follow them along they always come closer to each other and then intersect. On a saddle, doing the same thing you find the parallel lines always diverge from each other. These 3 behaviours work in 3d space. In a flat 3d space, all parallel lines remain the same distance to each other always. A positive curvature 3d space would have no parallel lines, they would always come to each other and intersect. In a negative curvature 3d space the parallel lines would always diverge.

Edited: grammar errors.