This is from the first episode of 日常 on Animelon, and was the answer to the question, "どうしたの？", from another character. I understand the individual parts: いろいろ == "various", やる気 == "motivation", and 出なくて is the negative Te-form of 出る, which I thought meant "to exit, go out", but how all that goes together to mean "I'm not motivated to do anything" is not obvious to me. Also, I'm also unclear on why the character ended her sentence with the negative Te-form in this case. Thanks!
やる気が出る is a phrase meaning "be motivated", you might as well remember it, along with things like やる気がない or やる気満々. In this case でる means something like "appear spontaneously", meaning 5 here: https://www.edrdg.org/cgi-bin/wwwjdic/wwwjdic?QMDJ%A4%C7%A4%EB. Here いろいろ means "for various reasons" or "in a lot of ways", but probably the translator thought it wasn't worth adding to the English translation.
Also, I'm also unclear on why the character ended her sentence with the negative Te-form in this case.
In this case the person is "leaving things up in the air", they have not really finished saying everything they want to say, and there might be more to come. So this "te" form is the continuative form where there is another sentence after it, but the person doesn't actually utter it.