To 'put the boot in' is British slang to kick someone, especially to give a vicious kicking to a helpless victim. The image is not of a karate kick, but kicking someone who has already been knocked to the ground.
It is also used metaphorically, to mean kicking a person when they're down, i.e., to make a bad situation worse.
This is the more modern version of the now mostly obsolete 'put the boots to'.