When the definition of the classifying space takes place within the homotopy category of CW complexes, existence theorems for universal bundles arise from Brown's representability theorem.
There exists an injection of G into a unitary group U(n) for n big enough.
On the other hand, the pull-back of the principal G-bundle P → M by the projection p : P ×G EG → M is also P × EG Since p is a fibration with contractible fibre EG, sections of p exist.
The total space of a universal bundle is usually written EG.
The idea, if G acts on the space X, is to consider instead the action on Y = X × EG, and corresponding quotient.