Geometrically speaking, if you draw a really, really big triangle between, say three galaxies, the angles of the triangle add to 180° in a flat universe. In a non-flat space, this would not be the case. For example, if you draw a triangle between, say, New York, Berlin and Rio de Janeiro on the surface of the Earth, the three angles between the lines would add to more than 180° since Earth is topologically a sphere and not flat. And if you draw three lines beween three points on a saddle shape like a Pringles chip, you'll find that the angles add up to less than 180°.
Fun fact: topologically speaking, no matter how you fold or bend a sheet of paper, it remains flat. A cilinder is a flat surface with zero curvature!