끼다 is fitting something snugly into place, so it is the verb for wearing gloves, rings and earphones rather than clothes: sharing a pair of earphones, splitting a pair of gloves one each, remembering to put it on. It also means squeezing into a gap, as in get in the middle, or tucking money between the pages of a book. With the helper for giving, it means letting someone into a group, which is what the thanks for including me line is about.