To tuck something in is to fold or hide it inside something else, often to keep it safe or make a neater appearance.
When you put on your favorite sweater and try to smooth out the loose threads, you might tuck them under the hem so they're not showing. This action of folding things under is also used for clothes with sleeves, like tucking shirts into pants or tucking in a collar to keep it from flapping around.
How common is "tuck"?
Word tuck is considered uncommon in modern English. It appears most frequently in fiction texts and less often in other writings.
Definitions
verb
(transitive) To pull or gather up (an item of fabric).
(transitive) To push into a snug position; to place somewhere safe, or handy, or somewhat hidden.
Example: Tuck in your shirt. I tucked in the sheet. He tucked the $10 bill into his shirt pocket.
(intransitive, often with "in" or "into") To eat; to consume.
(ergative) To fit neatly.
Example: The sofa tucks nicely into that corner.
To curl into a ball; to fold up and hold one's legs.
Example: The diver tucked, flipped, and opened up at the last moment.
To sew folds; to make a tuck or tucks in.
Example: to tuck a dress
To full, as cloth.
(ambitransitive, LGBTQ) Of a drag queen, trans woman, etc., to conceal one's penis and testicles, as with a gaff or by fastening them down with adhesive tape.
Example: Honey, have you tucked today? We don't wanna see anything nasty down there.
(music) To keep the thumb in position while moving the rest of the hand over it to continue playing piano keys that are outside the thumb (when playing scales).
(aviation) Ellipsis of Mach tuck.
Example: Never take a first-generation Learjet past Mach 0.82; it'll tuck hard nose-down and you won't be able to pull out from the dive.