出る literally means to leave or to go out.
In a conversation I heard it being used in "ちょっと手が離さないから、出てくれる" a request to check the door after the bell rang. Without knowing the context I would interpret this as "can you get out for me?", but this is clearly not the case.
Does 出る have other meanings / does the meaning of 出る change when used in an expression?
Firstly, it should be 「[手]{て}が[離]{はな}せない」 instead of 「手が離さない」. It should be in the negative potential form.
Secondly, 「[出]{で}る」 does not always mean "to leave" or "to go out" and that is exactly the origin of your misunderstanding this time.
You can say:
「[電話]{でんわ}に出る」 = "to answer the phone" ← This is close to the usage in your sentence.
「テレビに出る」 = "to appear on TV" ← Very different from "to leave", right?
Here, 「出る」 means "to reply (to the bell that rang)".
Thus, 「出てくれる？」 means "Can you get that?", with "that" referring to the bell. It must be said with a rising intonation towards the end.
"I'm busy now: Can you get that (for me)?" The speaker is asking the other person to go to the door to see who it is (and do whatever is needed to be done).
