It is slightly easier to evaluate If we are able to calculate this for all directions of than we can find the previous integral. The dot-product can easily be written in Carthesian coordinates. Since the eigenstates contain spherical harmonics we would like to write the dot-product with spherical harmonics. The radial integral involves the radial parts of the wave-functions. This depends on the potential and hence on the type of atom.
Different atoms have different transition rates because their radial wave functions are different.
The angular integral just contains spherical harmonics, and is the same for all atoms. The easy part of the angular integral is the azimuthal part, or the phi-integration. But m and m' are given by our choice of initial and final state, and M has to match the difference. The theta integral is more complicated. That is what we are doing here.