In the comments, the OP says that he is willing to restrict to $G$ a compact group, but not $G$ a compact Lie group. In this generality, the claim is FALSE.

A counterexample is the group of $p$-adic integers. For this $G$, the classifying space $BG$ fails to be locally contractible, because it contains a copy of the Hawaiian earrings in it.

The comments already point out that, for any compact topological group $G$, the classifying space $BG$ is paracompact. As the comments point out, if you drop the assumption that $G$ is compact, paracompactness of $BG$ can fail. The statement the OP wants ($BG$ paracompact and locally contractible) is true for compact Lie groups, however, as observed in the comments.

In the comments, the OP says that he is willing to restrict to $G$ a compact group, but not $G$ a compact Lie group. In this generality, the claim is FALSE.

A counterexample is the group of $p$-adic integers. For this $G$, the classifying space $BG$ fails to be locally contractible, because it contains a copy of the Hawaiian earring in it.

The comments already point out that, for any compact topological group $G$, the classifying space $BG$ is paracompact. As the comments point out, if you drop the assumption that $G$ is compact, paracompactness of $BG$ can fail. The statement the OP wants ($BG$ paracompact and locally contractible) is true for compact Lie groups, however, as observed in the comments.