도장 is "stamp". A stamp was much more widely used in Korea than in the west. In the old days everyone had a stamp of his own and used it in place of a signature. 손도장 is a palm print. Sometimes one makes a print of his palm because he forgot to bring his 도장. 발도장 is an extended concept of 손도장. It means the print of one's foot. But it is distinguished from 발자국 (footprint). 발자국 is the common word for footprint. 발도장 is something you leave like your 손도장 or 도장 (sometimes used as a joke). 도장을 찍다 is the common expression for "stamping", or "taking the print (onto a paper)". So 발도장 꾸~~~욱 찍고가요 is a lighthearted remark said half jokingly, not meaning much. It's like "Well, I've been here, and my footprint is left firmly(꾸---욱)". It's just saying "I've been here" in a more interesting way. 꾸----욱 describes the way something is printed or pressed with deliberately force.