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.