I think this is a pretty good intuitive introduction. I think there’s a bit of muddling between the fundamental group and first homology group with the article: the algebra of the fundamental group where loops are merged together cares about the order you do each loop and I don’t think it’s intuitive that a + b = b + a for the torus, indeed I found this quite unintuitive when I learned it. Note that the group isn’t a…
Although you are right about the plane and sphere, a much earlier proof of them being not homeomorphic is obtained by simply stating one is compact while the other is not.