I think it's easier if we break this down first. 손 is hand, 내(밀다) is to reach out, to stick out, to extend, "ㄴ" of "민" makes the 내밀다 verb an adjective, and 건 is a speech abbrev. for 것 (thing)은, which sounds like "거는" in speech. It would be rather difficult to interpret the exact use of the phrase "손내민건" without its preceding and succeeding words/phrases, but it sounds like it can possibly be translated into 손 내민 이유는, which can be translated into "a reason as to why (I/you/she/he/it) extend (my/your/her/his/its) hand." For example, 내가 그 순간에 "손내민건" 그 사람을 돕고 싶었기 때문이였다. "The reason I extend my hand to him was because I wanted to help him."