This seems to follow from the fact that the Chern classes are integral, an observation which is more mysterious using some definitions than others. For instance, I know of no direct way to prove that the Chern classes defined using Chern-Weil theory are integral; I would be very pleased if someone knows a way.
Using other definitions it is at least more standard; see the answers to this question, for instance: Cohomology ring of BG.
