Here’s one good reason that only requires high school geometry. Remember Euclid’s first axiom: given any two distinct points, there exists exactly one line that contains both of them. If you don’t identify antipodal points on the sphere, then this axiom fails. “Lines” on the sphere are great circles, but, for example, there are infinitely many great circles containing the north and south poles. It turns out that this is fixed once you identify antipodal points. Of course, not all geometric spaces need to satisfy Euclid’s axioms, but this is a desirable property for a space that we want to have some sort of linear structure.