What does よろしく Yoroshiku mean?

Yoroshiku asks for good treatment in future. It is said at introductions, which is why English subtitles render it as nice to meet you, and that translation faces the wrong direction.

What it is built from

The adjective 宜しい (yoroshii) means good, fine, acceptable. In this adverbial form the sense is roughly do well by me, or handle this favourably, with the verb left out because it does not need saying.

Left-out verbs are ordinary in polite Japanese. The request is complete without one, and stating it would make the whole thing heavier and stranger.

Backwards and forwards

Nice to meet you comments on an event that has just occurred: we have met, and it was pleasant. The Japanese says nothing about the meeting. It concerns the relationship that starts now and asks the other person to be good to it.

That is why the same word closes an email requesting something, ends a message handing over a task, and accompanies a favour being asked. None of those situations involve meeting anyone.

The politeness it performs

It puts the speaker in the other person's hands. Whatever comes next, you are being asked to handle it well, and the speaker has acknowledged that they are dependent on that.

Which connects it to sumimasen and to amae: all three organise politeness around the acknowledgement of dependence rather than around gratitude or apology as English understands them.

What English has instead

Fragments. Looking forward to working with you covers the professional case and is stiff. Thanks in advance covers the request case and can read as presumptuous. Be gentle with me is a joke.

The reason the fragments do not combine is that English politeness is mostly retrospective, thanking for what was done, or hypothetical, asking whether someone would mind. A standard formula for committing yourself to someone's future goodwill has no obvious slot, which is exactly the hole the word fills.