I think the point that was confusing me/missing link was that spherical harmonics functions are the solution of the Laplace's differential equation:
Orthogonal means the functions "pull in different directions". Like in linear algebra, orthogonal vectors "pull" in completely "distinct" directions in n-space, it turns out that orthogonal functions "help you reach completely distinct values", where the resultant value (sum of functions) is again a function.
SH are based on the associated Legendre polynomials, (which are a tad more funky than Legendre polynomials, namely each band has more distinct functions defined for it for the associated ones.)
The Legendre polynomials themselves, like SH, are orthogonal functions. So if you take any 2 functions from the Legendre polynomial set, they're going to be orthogonal to each other (integral on -1..1 is 0), and, if you add scaled copies of one to the other, you're going to be able to reach an entirely distinct set of functions/values than you could with just one of those basis functions alone.
Now the sphere comes from the idea that, SH functions, use the Legendre polynomials (but Legendre polynomials are 1D functions), and the specification of spherical harmonics is __a function value for every $\phi \theta$. There is no "sphere" perse.. its like if you say "there is a value for every point on the unit circle", it means you trace a circle around the origin and give each point a value.
What is meant is every point on a unit sphere has a numeric value. If we associate a color to every point on the sphere, you get a visualization like this:
This page shows a visualization where the values of the SH function are used to MORPH THE SPHERE (which is part of what was confusing me earlier). But just because a function has values for every point on the sphere doesn't mean there is a sphere.