Comparison of rate of reaction in bromination and chlorination
According to the Hammond postulate, a highly exothermic reaction is more likely to have a low transition state barrier than a less exothermic reaction.

Lower barriers lead to faster reactions.

This is a multistep mechanism, so for this to be relevant the rate-determining step.  According to this lecture the rate limiting step is the hydrogen abstraction from the alkane, forming a hydrogen-halide bond.

The bond dissociation energy for HCl is 431.8 kJ/mol and 365.7 kJ/mol for H-Br.  The formation of the H-Cl is this much more exothermic, and the reaction will be faster.