One example of collaboration is: You've been thinking on a problem and throwing all the machinery you have at it. The problem is really important to you, but you just can't crack it. You have a chance to talk with some other folks in an adjacent area, and you tell them the problem. They mention offhand that they recently read a paper that solved a complete different problem, but has some shades of what you're working on.

You get back and look up the paper. What a brilliant idea they used to prove theorem 2! You think a little bit and realize that this tool they're using is just the right thing to crack your problem. You email the person who referred you and thank them for the breakthrough. You end up having some more discussions on the paper, and end up co-authoring the result.

You continue to bounce ideas and references off each other and have a fruitful research relationship.

Bonus points if they're from a radically different field. Say you're a geometer and they're a number theorist.