'A rocket lifts off because the exhaust gases exert a force on the rocket'.
However this statement confuses me as I would like more detail. How does it actually work? What are the forces and where do they come from? How could I build an equation to see it more clearly?
Imagine you have cube shaped box filled with pressurised gas. The gas pushes against all the walls. The resulting force from the gas on the box is zero because the forces on opposing walls cancel out.
And now imagine you remove one of the walls but still maintain the pressure inside. Two things will happen: (1) the net force is no longer zero - the gas still pushes against the wall opposite the one you removed but there is no counter acting force. This is where the thrust comes from. (2) the gas rushes out through the opening you made - these are the exhaust gases that exit the rocket engine.
To prevent the pressure inside the box from dropping you have to keep re-supplying the pressurised gas - this is done by injecting fuel and oxidiser into the box (engine) and burning it. You push in relatively small volume of liquid (or gas) at high pressure (you have to push against the pressurised gas that is already in the box). As it burns it expands so you end up with a large volume of pressurised gas in the chamber such that the pressure inside does not drop despite the large flow out.
Basic principle is simple, it works because of action-reaction force pair, like here in the picture,
You throw boxes out of car with force, $$ \tag 1 F_{\text{thrust}} = \dot m v_e, $$
where $v_e$ is exhaust velocity and $\dot m$ is car mass change over time in $[kg/s]$ and same force exhaust applies to your car due to Third Newton law. By analogy $\dot m$ measures how much such people you have standing on a trolley. The more you have,- the more boxes they can throw out per time unit,- each with momentum $m_{box}v_e$ and as such total force passed back to trolley scales-up linearly. But core rocket moving principle is as primitive as gun recoil.
The difference being is that in rocket case we favor recoil and try to maximize it by any means, for example by choosing favorable nozzle which accelerates gas flow, since full rocket thrust equation is, $$ \tag 2 F_{rocket} = \dot m v_e + \Delta pA_e ,$$
where $\Delta p $ is pressure difference between nozzle input and output, and $A_e$- nozzle exit cross-section. So, in conclusion you can call rocket "a maximized recoil system."
