If someone goes out on a limb, they take a risk. They say or do something that could fail completely or be harmful. If you find yourself out on a limb — in a dangerous position — be very careful!
A tree's "limb" is a branch. Imagine climbing a tree to free a kite stuck there: you crawl far out on a high branch to reach it, wondering if the branch will hold you. That’s the image behind out on a limb. You’re going out on a limb if you buy an expensive gift you can’t return, without knowing if your friend will like it. Sometimes, if you're the only one daring to disagree with the majority, you may find yourself out on a limb.