How did Han Solo and Chewbacca hook up?

We don't know. There's a whole, quite involved backstory in the expanded universe (see Erik Blythe's answer) but that's no longer canonical - it's a 'Legend' now. A canon source may bring that story back into the fold or fill the gaps in some other way. read more

Chewbacca owes Han Solo a life debt. This in itself seems like a very important ritual and obligation to a Wookie. Until the stand alone Han Solo movie hits the screen, we have to wait and speculate. read more

In short: Chewbacca was a slave under Empire (like many Wookiees) and Han saved his life from another Imperial officer (twice). Among the reasons for that was that earlier in life another Wookiee (Dewlanna) saved Han's life at the cost of her own. read more

Also Read: 22 Times 'Solo' Recycled Moments From the Original 'Star Wars' Trilogy. As Han dodges the fighters, he tells Chewie to deploy the Falcon’s landing gear. Han explains that he’s about to pull a move he learned from his pal Needles, the best street drag racer on Corellia. read more