Mechanism for the ring contraction
As has already been said in the comments, this is a free radical-based reaction. The peroxide $\ce{R1-O-O-R2}$ acts as a radical initiator( where $\ce{R1}$ and $\ce{R2}$ represent alkyl groups). These peroxide radicals generate $\ce{^.CCl3}$ which attacks the double bond and completes the reaction as shown.

An important point to note is that $\ce{^.CCl3}$ is regenerated at the end of the reaction, so it can be described as a catalytic intermediate for this mechanism