Steve's answer is not bad but here's another way to understand the phrase... Unlike the fish who doesn't realize he is about to be hooked when he bites, the person who says "I'll bite" is acknowledging that the first speaker has thrown out "conversational bait" and is setting up the listener in some way (for a joke, a trick, an insult, whatever). The second person says "I'll bite" indicating that he knows that he is being set up but is willing to play along, thus inviting the first person to continue with his joke. "I'll bite" is also used to suggest that what was said by the first person is not quite believable. In this case, "I'll bite" indicates that the speaker is willing to suspend disbelief long enough to hear what the first speaker has to say but that he reserves the right to dismiss the whole thing as rubbish or poppycock.

This is a fishing metaphor. When you fish, you put bait on your hook, and wait for the fish to bite. The fish bites not realizing there is more there than just the food, or perhaps, realzing there is, but being willing to take the risk. From there, you have the fun of reeling the fish in. In this metaphor, someone puts a tantalizing and incomplete statement out in front of people waiting for a reaction - that's the bait. When someone can't resist, and asks about it, they bite the bait, and then realize there is more to the statement than what was first put out.. In this case, an idea is put out and the speaker waits. Someone can't leave it alone, so realizing there is probably more to it than they see, and getting ready for the surprise, they may preface it with, "I'll bite." It's biting like a fish bites the bait when you go fishing.