It's all those exhaust gases emerging from the nozzles at super high speed that the rocket engines are emitting. At the back of the combustion chambers there is a high pressure that pushes up against the inside, raising the rocket, whilst the same pressure is pushing the gases out at the bottom. The momentum of all that high speed gas is equal to the momentum gained by the rocket.
At the start of the flight, the whole process is INCREDIBLY inefficient because there's a lot of force but not much distance raised. It gets better as the rocket starts to go faster.
PS you don't need the ground to be there for the gases to 'push against'. However, it's got to be true that, (as with a hovercraft) whilst on the ground, there will be a bit of excess pressure under the rocket, which will help to lift it the first meter or so. Above that, the gases can escape so easily that the pressure is not helped by the presence of the ground.