I want to provide a bit of intuition about how I think about it (I believe there is a formal proof of the classification along these lines)the following lines.
Being simply connected means that whenever you have curve, you can always get a disk inside. If there is more than one way to glue a disk, you must have a Riemann sphere. If there is always exactly one way, you can take the whole picture and put it step-by-step on a complex plane. Once there, you have either the whole plane or you're inside the complement of a ray. In the latter case, you are between a disk and a disk, so there's some approximation thing that says you're also a disk.
The three cases can be distinguished by their global symmetry group: there are three different constant curvature metrics, with the curvature resp. 1, 0, -1 for the sphere, plane, and disk case. Therefore you have three slightly different symmetry groups .which can be written as the isometries of quadratic forms in 3 dimensions: $SO(x^2 + y^2 + z^2)$, $SO(x^2+y^2)$, $SO(x^2+y^2-z^2)$.