Space isn't flat in the sense that it is flat like a piece of paper (i.e., it only has two dimensions), space is flat (maybe? we're not %100 sure) in the sense that it isn't a closed curve.

Think about it this way. On a curved object, like the surface of the earth, you can walk along the equator, turn 90 degrees, walk to the north pole, turn 90 degrees again, walk down to the equator, and turn 90 degrees again - tracing out a triangle with 90 degree corners. But that's not standard "flat" geometry, because triangles in that system can't have all 90 degree corners - it only works in a curved, closed system like a globe. On a curved coordinate system, parallel lines - like lines of longitude - eventually meet. (They all converge at the north and south poles.)

The universe could be like this. It could be a closed system that curves in one direction or another. But some data suggests that this isn't the case and parallel lines in space will never meet, just like they don't in two dimensional "flat" geometry.