The meaning of the Hungarian sentence is, "The young man starts moving toward the woman."
Although "el" is often translated as "away," in this case, the directional meaning is not appropriate. There's nothing in particular that he's moving away from. The function of the preverb particle "el" here is aspect: it indicates that the focus is on the start of the action. This can happen with a lot of the other preverbal particles that you meet in the lessons here: they can indicate direction, but when they're not doing that, they may be playing an aspect role.
Great that makes perfect sense-i've seen how aspect can work before in other examples but couldn't quite figure out how it was relevant here. So without the -el, the sentence would mean something like, "the young man is moving towards the woman", i.e. the process is already underway. I don't think the Duolingo English translation really encapsulates your translation well though. It's all very well to say "I'm leaving for the train now" or "I'm leaving for Austria tomorrow" but "to leave for the woman" doesn't sound quite right.
------- the young man leaves for the woman implies that he quits his job in order to go with, or to , the woman , that something impacts his life so that he must go away on her account. it's not (the way the hungarian is written ) that he is simply going from here to some (short ) distance away up to a woman . . .
Big 14 jan 18
What is the Hungarian sentence trying to describe here? The English translation sounds a little awkward. Is it "The young man is leaving to go and see a woman" i.e. leaving a room or building and travelling a distance to reach the woman or is the woman closer by e.g. "The young man leaves (the current people he is with) to go towards the woman (e.g on the other side of the room) ?
Also what function does the "el" play here i.e. what would "A fiatal férfi indue a nőhöz" mean?
"A fiatal férfi indul a nőhöz" means nealry the same. "indul" and "elindul" is mostly interchangable in everyday Hungarian. The only big difference that pops up to my mind at the moment is the case of finished action. Such as "I have left" is "Elindultam" and not "Indultam".
Everything else you and jsiehler said previously is also correct. :) The original Hungarian sentence, as jsiehler translated means the young man moves/goes/travels towards to woman.