The direct product is a way of determining how objects of different symmetry can interact. Quantum mechanical processes are often a coupling of two disparate things. For exmaple, light can couple two states of different symmetry. Vibrations can cause mixing of different electronic configurations. Magnetic fields and spin-orbit interactions can result in state mixing. The direct product is the way to determine the symmetry required for mixing. If we take the direct product of two objects of different symmetry we can determine the irrep of the interaction needed to couple them.
Frequently systems get so complex that it is difficult to organize all of the necessary effects and possibilities. The product table permis rapid determineation of the relevant interactions in order to scan for possible vibrations, polarizations and so on that can give rise to coupling.
