It's a strong affirmation, rather colloquial, I think. It can be used on any yes-no questions to show agreement or the intention to certainly do something. "Will you go to the concert tomorrow?" - "You bet!" (Of course, I am. I've been looking forward to it for half a year.)

you bet = yes = absolutely

Thekyr is absolutely correct with her explanation. However, "you bet" is also often used in the USA as another way to say, "OK." So if someone says, "I'll see you later," you could reply, "You bet." The stronger you say/inflect it, the stronger the meaning.