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.