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A quasi-star is in simple terms a luminous star with the black hole as the core, and the radiation from the black hole is pushing the things away and also having an equilibrium while it eats up the stellar material!
These are formed generally at a protostar system, with 1000 solar masses. The protostellar systems may have formed from high-density stellar dust collections or by dark matter halos, gasping in all the gases. The protostar system is formed in the early days of any star formation and collection of dust and gases is constantly done and it increases in size. Once in a suitable size form, the core of the protostar can collapse into a black hole at the core. The outburst in the form of supernovae is contained near the protostar itself (assuming its large size, which can theoretically span up to a galaxy itself).
The black hole in the core of the star then keeps taking in the dust and matter nearby the core of the star and emits radiations that keep the outer layers intact from the strong gravitational pull of the black hole. Equilibrium is maintained as the protostar keeps collecting the gases and forming layers as the black holes keep emanating radiation while they consume the mass.
The typical life span of such a system can be seven million years which is, again, very small for cosmological scale events. These protostars and quasi-star systems can generally be observed in the early days of the Universe.
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