NMR 8 - The origin of spin coupling

Consider a simple case of two CHs next to eachother:

The magnetic shielding of H_A is slightly affected by the spin of H_B. In some molecules, H_B will have a +½ spin, in others it will have -½. At room temperature, spin populations are pretty much populated 50:50. So half of H_A will be in one environment, and half of H_A will be in a slightly different environment.

Using the same logic for H_B, the spectra of these two hydrogens will be a doublet of doublets.

The n+1 rule just sums up the result of this thinking process. For example, for H_A coupled with two H_Bs, it gives a splitting of three. The three different H_A environments are both H_B at +½ spin, both at -½ spin, and one at +½ and the other at -½.