'kicking in' means starting. The new rules kick in on Thursday. The new rules start/take effect on Thursday. The heating hasn't kicked in yet. The heating has not yet started to warm the place. The drugs will kick in any minute. The drug effects will be felt very soon.

This means that the person who ingested drugs will feel their effect soon.

Hi JamesR, Thanks for your comment it really helped me understand the meaning.

They mean that the effects of the drug (illegal or legal) will be felt momentarily.