A decent response on how the burning process works but how does this glowy burner push the spacecraft
I think it's the near-invisibility of the exhaust from second stage engines in videos (compared to the first stages) that has caught your attention.
It always bothers me too, but for one or more reasons involving the fuel/oxidizer ratios (mixture) for 2nd stage operation and the choices of fuel itself and also because there's no atmospheric oxygen to facilitate post-burning (think carbon soot particles as glowing embers or bits of ash in final combustion) in space.
In fact, if you watch a first stage launch all the way up, especially from a camera mounted ON the rocket, you'll see an amazing evolution of the plume, it spreads out and can get pretty dim.
So like the comments say, there's no difference in propulsion between first and second stages - both push out a HUGE mass flow of very fast exhaust gas, and it's the reaction from the exhaust mass going one direction that propels the spacecraft in the other direction.
It's just as powerful whether it's blindingly bright and opaque, or its nearly transparent and invisible.