Imagine a car factory that has a number of sub-assembly lines. One makes brakes, another engines, another transmissions, another alternators and so on. We know a complete car needs these in set ratios to make cars, so that is the ratio they are made in. Imagine a factory that made huge numbers of brakes, transmissions, alternators, but a limited number of engines. Engines are the limiting factor = they limit the number of cars that can be made - all the other parts will fill all the factory space until the walls break.
A virus has to operate in a similar way = translates the infecting RNA and starts to make all the viral components that make up the virus. The cell makes functional viruses until the cell walls break - triggered by fullness or exhaustion of cellular resources that trigger cellular rupture (apoptosis) and as many complete viruses that were completed enter the circulation to infect other cells. All the extra unused parts are released. Note all parts in all stages are also displayed on the external cell membrane as immune system cues. This is why a true infection produces a very broad antigenic response compared to the response to spike alone. This also explains the 'super immunity' from vaccinations coupled with infection - in any order, as we have observed and has been reported.
The omicron mutation seems to have optimised the internal cellular production of a larger number of components in the (more or less) optimal numbers needed to maximise the number of completed virus particles manufactured before cellular break-out. All the components are still displayed = antigenic cues to the immune system are still displayed = good immune response. This process, as a whole, makes a larger number of infective particles released in a shorter period of time.
This explains the faster spread via more particles and also explains why the infection 'breaks out'.
It breaks out even when vaccinated because the adaptive immune system takes about 5-7 days(varies) to increase it's antigenic response and the Omicron cycle only takes 3-4 days to start making large numbers of infective particles. This means it can ramp the increase of infective particles ahead of the antigenic response. This also explains why the break out infection starts to decline in that period after ~~5-7 days as the adaptive immune system gets the upper hand and the infection subsides.