Taylor expansion in Lattice Boltzmann Method derivation

Currently I’m trying to understand Lattice Boltzmann Method for solving CFD problems. In its derivation BGK approximation is used to get rid of complicated collision integral. But when they come to descritization of equilibrium function (Maxwellian). they use Taylor expansion up to the second order of velocity, which is valid for low Mach numbers and constant temperature. So my question is: why do we use this expansion? why original expression of equilibrium function is not used? And where does condition Ma≪1 come from? Do we need the condition just for expansion or it is needed somewhere else?

Thanks in advance!