It’s been basically covered before by those before me, but something that people left out was the fact that Han was enslaved to a smuggler in his youth. He had a crew maintaining his ship, and the chef, a female Wookie, went out of her way to care and raise Han as best as she could. If I remember correctly, she ended…

In the old EU, Han was an orphan and in his adolescence he was basically enslaved by a smuggler and made to do menial tasks to keep his ship clean. The person who always looked out for him was the chef, a female Wookie who raised him for the most part, hence why he understands the language.