OK, there are LOTS of reasons for this. Reason #1, the cardinality of the reals is the cardinality of the reals squared. Thus, since the surface of a sphere cannot be countable, it must be at least the cardinality of the reals.
Reason #2. Complex analysis. The complex numbers can be expressed as numbers on a plane (real and imaginary axes). The Riemann sphere is used in complex analysis by the projection that is in reason #3.
Reason #3. Put the south pole of a unit sphere on the origin of the plane. Consider a line segment between the north pole and the complex plane. That line segment intersects the sphere in one and only one place. In fact, rays eminating from the north pole that intersect the sphere, do so in only one point and also intersect the plane in only one point.
There are more reasons, but I would need LaTeX and this is, after all, a political forum.
The chemist saw the fire and thought A fire needs oxygen. He grabbed a blanket, smothered the fire, and went back to bed.
The physicist saw the fire and thought A fire needs heat. He grabbed a pitcher of water, poured it on the fire, and went back to bed.
The mathematician saw the fire, looked around the room, and saw a fire extinguisher. He thought A solution exists and went back to bed.