Traditionally the explanation involves distinguishing between nature ("what") and person ("who"), which is the basis of the term "hypostatic union". There is one Person (which the gospel of st John refers to as "the Word"), which is of divine nature, i.e., is God; this Person assumed also a human nature, on the incarnation; and that is Jesus. This is what is meant by Jesus being truly God and truly Man: the one Person had united in himself both natures, the divine and the human.

I agree it's unfortunate that these kinds of questions sometimes get answered by inadequate metaphors or simply by dismissals. The whole joy of theology, while still requiring faith, is trying to answer these questions rationally.