When something "kicks in", it means that it starts to work. This is a common expression for talking about medicine:

The painkillers haven't kicked in yet.

Something else that can "kick in" is an instinct, a feeling, or a certain way of thinking:

As soon as he heard the gunshot, his military training kicked in and he dropped straight to the ground.

And something else that "kicks in" is a retirement plan or some other kind of payment that you're scheduled to receive at a certain time:

When she turns 21, her trust fund kicks in and she won't need to work any more.

A more formal phrase with a similar meaning to "kick in" is "take effect":

The doctor said that it would take a few days for the antibiotics to take effect.