However, since the mouth is open, the air has somewhere to go. The air pushes out the mouth of the balloon. Since the air is now moving, the rocket balloon also moves. This follows both Newton's Third Law of Motion ("for every action there's an equal and opposite reaction") and the law of conservation of momentum.
There is friction between the string and the straw, but the straw is made of material with a low coefficient of friction (because it's smooth) and the weight of the balloon and straw is light, so the overall frictional force is effectively negligible.
This is essentially the same principle upon which rockets work. They burn fuel to propel matter out the back forcefully. The rocket moves in response to this. Since a rocket typically weighs much more than a balloon, of course, it takes more propellant to get it moving.