면서 주다

hand it over while saying something

"-(으)면서" is "while, at the same time as", and 주다 is giving. It describes what accompanies the handover: offering a sweet while saying "this is famous back home, want to try?", he gave it to me while dropping off the luggage, I'll pay you on the way out. It also covers the spirit of the giving — a gift is something you hand over hoping they'll get good use of it. The accompanying words are often the whole point of the line.