Because for a sufficiently small sphere (for example), I can wrap a sheet of 8.5"x11" (or A4, for those using the metric standard) completely around it. Now you would agree that a sheet of paper has a finite surface area, so why wouldn't a sphere also have a finite surface area? If a sphere had an infinite surface area, it would take an infinitely large sheet of paper (or an infinite amount of paint) to cover it. Yet you can travel an infinite distance on the surface of the finite sphere (granted you will be traveling over the same points at some point).
Topography (the study of surfaces) makes my brain hurt.