Before we get into the science of rocket launching, let's take a look at some rocket launches:
See the fire blasting out of the nozzles of the rockets? By creating a force downwards (blasting the flames towards the ground), the rocket can fly upwards. This is because of Newton's Third Law of Motion.
Newton's Third Law of Motion states that for every action (a force), there will be an equal and opposite reaction.
For example, if you kick a ball into a wall, the ball will bounce back. That's because when the flying ball applies a force on the wall, and the wall gives the ball a reaction force.
The jet of fire blasting out of the nozzles act as an action towards the ground. An opposite reaction occurs, creating a force that pushes the rocket upwards. Blastoff!
In a liquid rocket, fuel and oxidizer are pumped into a combustion chamber where they are mixed and burned. The combustion produces great amounts of exhaust gas at high temperature and pressure. The hot exhaust is passed through a nozzle which accelerates the flow. Therefore, thrust is produced!