And more precisely, how did Chewbacca come to owe life debt to Han Solo?
I'm also trying to find the first source that says Chewbacca owes a life debt to Solo. I thought it was from the movies, but I cannot find the exact quote.
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To answer your second question first (first published mention of life debt):
The first mention of the idea that a debt exists is in Episode V (ESB), when the debt was extended to Leia (Han telling Chewie to now take care of her). This is open to argument, but I personally agree with the concept.
However, "life debt" as a term it isn't present in the SW movie trilogy.
The first explicit mention of the term is in Brian Daley's "Han Solo Adventures" book trilogy - but NOT in the context of his debt to Han!
"What do we owe Badure?" Han hollered back. "He made a business offer, Chewie. " But he knew better. Wookiees will honor a Life-Debt over anything else; he'll never walk away from it, Han thought. Chewbacca growled another angry comment.
So far I wasn't able to find any mention of life debt as a term in Han/Chewie context that predates A.C. Crispin's [Han Solo trilogy] - see below.
The origin and details of the debt is from the books and mostly comics.
Some parts of it were in "The Hutt Gambit" (second book in A.C. Crispin's Han Solo trilogy);
The original mention of the slaver story is in 1978's "The Wookiee Storybook", though that doesn't mention life debt yet.
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.
More details on SW Wikia:
Chewbacca often attempted freeing Wookiees slaves ... On one such endeavor, he became involved in a botched slave transfer between a Trandoshan crew and an Imperial commander Nyklas. Chewbacca orchestrated the escape of close to all of the slaves which consisted mostly of Wookiee children, but in the process had his ship come under fire by Imperial TIE Fighters led by Lieutenant Han Solo. Nyklas was infuriated at the loss of the Wookiee slaves and ordered Solo to kill the Wookiee (who had been rendered unconscious by the firefight). Solo refused, inciting Nyklas's fury, but the commander eventually calmed enough to realize Chewie's value as a slave. .
... Nyklas worked Chewie unmercifully, taking great pleasure in whipping the demoralized Wookiee. However, Nyklas did not count on any more interventions by Solo, and the young lieutenant saved Chewbacca from being whipped to death. The two escaped from Imperial influence with a small bounty on their heads, and Solo was drummed out of Imperial service.