"Flat" in this context means in terms of curvature. Using 2D as an example you can have a piece of paper which is 2D and flat or something like the surface of a balloon which is 2D and curved.

The problem is that, from the perspective of any beings that live on and are constrained by those 2D surfaces, the world just looks "flat" to them in both cases because any 2D beams of light are also constrained to the surface. The balloon case is curved, but it is curved through a third dimension which 2D beings cannot perceive.

Stepping back up into our 3D work, there is an open question as to whether our 3D space is "flat" or "curved" in the 3D sense. If it was curved, it would be curved through a fourth dimension which we cannot directly perceive, so how could we tell?

Stepping back down into 2D, our 2D beings could indirectly determine the curvature of their world through triangles. In the flat 2D world, any triangles they made would have angles whose sum always equals 180 degrees. But in the curved 2D world, you would be able to make triangles whose angles sum to greater than 180 degrees.

This property also works in 3D. If our universe is flat, then triangles all have angles that sum up to 180 degrees and if it is curved then they could sum up to greater than 180 degrees. By picking distant objects (such as far away stars and galaxies) and measuring the relative distances between those objects, we can calculate their angles. Within a certain margin of error, we've calculate that our universe is either flat or has a very very very very small amount of curvature.