Kid gloves are gloves made of fine kid leather. Kid leather is traditionally made from the skin of young goats. It is very soft. Expensive baby shoes are made of kid leather, for example. "To handle/treat someone with kid gloves" means to deal with that person very tactfully or carefully. "The kid gloves are off" means that tact and diplomacy are discarded. The people concerned will treat each other very roughly.

"Kids Gloves are off'' may refer to the idea that the innocence or 'beginner level' of combat (as it were), is no longer apparent. Hopefully that makes sense?

It is a little puzzling. It is not a standard expression. It is a combination of two different idioms. 1) "The gloves are off," literally refers to a boxing match. In an athletic contest, the boxers wear gloves to reduce the injuries to each other. When "the gloves are off," it means a brutal fight, without protection and without rules. 2) A baby cow is a "kid." Kid gloves are soft gloves made of calfskin. In certain kinds of delicate work, the worker will "handle it with kid gloves" to keep the pieces clean and avoid scratching them. By extension, "to handle something with kid gloves" means "to be very careful." In dealing with a difficult person, someone might say "he has a very hot temper; you need to handle him with kid gloves." The combination clearly suggests that the people involved are not going to be considerate, tactful, or careful with each other. They are going to be angry, direct, impolite, and willing to hurt each others' feelings.