Great question! When you blow up a balloon, the balloon stretches. Sometimes it can be pretty hard to blow up a balloon, because it doesn't really want to stretch. And if you don't tie it off right away, it will probably unstretch by pushing air out of the hole. When the air comes out of the hole, it creates what's called a force. (You can think of a force as something that pushes. You can tell that this happens because if you let the air out of the balloon slowly and hold your hand in front of it, you can feel the air pushing on your hand.)
Well, one of the most important rules about the way the world works is that "for every force there is an equal and opposite force." Basically what that means is that if you're pushing on something, it has to be pushing back on you. (For example, if you lean on the wall, the reason that you don't fall through it is because it's pushing back on you.) Another way to think of that is that if something pushes (applies a force) in one direction, then it will be pushed (feel a force) in the other direction.