The title to this article seems simple enough. And everyone gives the same answer: Because of the fire which comes out of the bottom of the rocket.
But if they are asked the next question -- "Why does the fire coming out of the bottom make the rocket go up?" -- most people draw a blank, or they have an odd misconception that the fire coming out of the bottom pushes against the ground to make the rocket go up.
But if you address the "pushing against the ground or pushing against the air" idea by asking, "Oh, so in space where there is no ground or air to push against rockets have no effect?," suddenly they are very puzzled because it dawns on them that they have no idea why rockets go up.
In fact, it's not the fire pushing down which makes the rocket go up, but rather the fire pushing up against the rocket which makes the rocket go up. The fire pushes up against the rocket at two places -- inside the combustion chamber, where the fuel explodes, and the explosion of the fuel pushes up against the top of the inside of the combustion chamber.
And the exploding fuel molecules coming out of the bottom of the rocket push up against the exhaust cone.
The idea is this: When the rocket fuel explodes, explodes, explodes in a constant stream of activity, the molecules in the explosion bounce around inside that combustion chamber with extraordinary violence. They bounce against the left side of the combustion chamber -- but because they bounce against the right side, too, those bounces cancel each other out and have no impact of rocket motion.
The firey, bouncing molecules also bounce against the TOP of the inside of the combustion chamber, very hard, and they would bounce against the bottom and cancel-out the bouncing against the top -- except that there is no bottom -- there is a hole, there. As a consequence, the upward push of the bouncing fuel explosion molecules is unopposed, and not canceled-out, and so those molecules succeed in pushing the rocket up.
The same hot molecules get to push the rocket up a second time, when they bounce against the sides of the rocket exhaust cone. When they pound sideways against the cone, because the cone is angled the molecules bounce down off the cone sides, pushing the rocket up again.
So, those two surface -- the top of the inside of the combustion chamber, and the angled sides of the exhaust cone -- are the two places whwere the rocket's fiery exhaust, even though it is coming DOWN out of the rocket, pushes UP.
What the firey exhaust "hits" outside the rocket does not affect rocket motion.