In fact, the energies in UV are exactly right to break the bonds in chlorine molecules to produce chlorine atoms. Notice also that it is quite acceptable to use a simple view of atomic structure. There is no point in using a complicated model of the atom if a simple one will do the job. Because we want to stress the fact that the chlorine atoms have single unpaired electrons, then we call them chlorine free radicals - or more usually just chlorine radicals. The name given to this is homolytic fission.
Reactions happen because things hit each other.
In this case, you need to think about what the chlorine radicals are likely to hit, and what could happen as a result of that collision. Let's start with the unproductive collisions. The least likely collision is between two chlorine radicals. If they do collide, they will combine to form a chlorine molecule. That's worse than useless because it removes the active free radicals from the system.