The 입 in 입이 걸다 means "mouth," but what does the 걸다 mean? In my dictionary, there is the verb 걸다, which means "to hang," and the adjective 걸다, which means "to be fertile." Since the phrase is 입이 걸다, not 입을 걸다, one can assume that the 걸다 in 입이 걸다 is the adjective 걸다 meaning "to be fertile." So, 입이 걸다 literally means "to have a fertile mouth."
The Korean word for "manure" is 거름, which probably comes from 걸다, the Korean adjective meaning "to be fertile." Farmers spread manure on land to make it more fertile. That would mean that 입이 걸다 could possibly be translated as "to have a manure mouth," which would be similar to the English phrase "to have a potty mouth." The word "potty" is another word for "toilet," which is where we deposit our "human manure."
Here are the two usage examples from my dictionary:
- 입이 건 사람 a foul-mouthed fellow
- 그는 입이 걸어 남의 욕을 잘한다. His mouth breeds ready slander.