I sometimes hear the phrase "잘 부탁드립니다" used in certain formal settings, for instance during introductions. It seems to mean, literally, just "I request (something)"; that doesn't really make sense to me. What is the actual meaning, and when is it appropriate to use it?
부탁 is literally translated to request, but more commonly do a favor.
어떤 일을 해 달라고 청하거나 맡기다
It is often used in introductions, but only when you WANT the listener to BENEFIT anyone in anyway.
When you are in a competition, you introduce yourself with 잘 부탁드립니다 since the audiences give you appreciation, or even can judge the winner.
When you meet someone whose experience is more than yours.
When you want the listener to take good care of, let's say, your son.
And a lot other situations.
It is spoken in reference to any relationship expected to last for some time, in which you, the speaker, will be dependent (however tenuously) on the kind disposition of the other, i.e. the addressee.
This dependence may be specific and significant or vague and marginal.
Examples on either end:
You are leaving a child with a teacher or your case with a lawyer.
You joined a weekend volunteer group and have to say a few words to a dozen people most of whom you don't know the names of.
As for its meaning, I agree with MujjinGun's adding '저를.' The phrase would not make sense unless you were putting yourself or your interest in someone's hands (again however tenuously). Thus, in terms of what it sounds like (since literal meaning you already know):
I put myself in your kind hands.
But this does not mean that the phrase is ever spoken with '저를' in it (it's not) or that the Korean speakers ever bother their head about what it means (no more than English speakers think about the meaning of 'Bless you!' when someone sneezed). The usage is conventional.
The expression can be used in informal situations too. For example, when you ask a favor to someone, you can say "잘 부탁해요" to wish them to take good care of the favor.
Generally, it means "please take good care of it". When used in introductions, "(저를) 잘 부탁드립니다", it means "please take good care of me".