"Tucking someone in" means sitting or standing next to their bed and helping them to get comfortable just before they go to sleep. Parents do this for their young children.

In this phrase, the word "tuck" means to straighten out the sheets and blankets on the bed. You can also "tuck in" a shirt, which means to put the bottom end of the shirt inside your pants or skirt:

Hey, you forgot to tuck in your shirt.