You don't need group theory to understand spherical harmonics. I think you're mixing up the concepts of spin and angular momentum with spherical harmonics.
If you want to understand spherical harmonics, you need a book on partial differential equations. Learn about various PDEs that have solutions, and the derivations of the various special functions such as Legendre Polynomials, Bessel functions, Hankel functions, and so also Spherical Harmonics. They're just the solution to the angular part of Laplace's equation in spherical coordinates. As such, they form a set of angularmomentum eigenfunctions in quantum mechanics the same way plane waves form the eigenfunctions for linear momentum.
That's really all there is to it.
