It has occurred to me that it is likely that the reason people are having such a hard time comprehending that Andy Stanley might not actually be criticizing or "downplaying" the virgin birth is that they see the opinion of Al Mohler as authoritative and that they have trusted him to provide a balanced and accurate summary of Stanley's message. The post immediately following this one will provide a transcript of the relevant portions of the sermon for your reference.
Let me address Mohler's points head on:
Stanley is speaking to people who may be quite skeptical of the story of the virgin birth because they don't understand that it is part of a much larger story of seemingly impossible and unbelievable promises that God has fulfilled throughout history. He is simply giving people space to hear the story for themselves instead of simply demanding that they believe it uncritically.
For someone to think that Stanley is characterizing the virgin birth of Jesus as truly unbelievable reveals that either they did not actually listen to the sermon past the first three minutes or so, or else they are deliberately mischaracterizing it.
That seems to be an obvious statement in light of the fact that two of the four Gospels don't mention it, Paul doesn't mention it, no sermons in the New Testament reference it, nor any New Testament writers other than Matthew and Luke. Clearly, it is not the "hinge" of the Christian faith.
Certainly, it is a hinge for some people's theology, but that is a reflection of their theology, not the New Testament.
Note how Mohler has shifted the question here:
Stanley claimed that "Christianity does not hinge on ... the stories around the birth of Jesus", and Mohler says that the Bible stories about Christ's incarnation are "the central truth claim of Christmas." Stanley is talking about Christianity and Mohler is talking about Christmas as it they are the same.
Mohler subtly misquotes Stanley and pulls his words out of context. Stanley actually said:
"...if somebody can predict their own death and then their own resurrection, I’m not all that concerned about how they got into the world, because the whole resurrection thing is so amazing, and in fact, you should know this, that Christianity doesn’t hinge on the truth or even the stories of the birth of Jesus."
Mohler claims that Stanley said, "it doesn't matter how he came into the world," when Stanley actually said, he was not all that concerned about it in light of the resurrection. If one can accept the resurrection of Jesus, then the virgin birth is easy to accept.
Mohler continues his thought to emphasize that the Bible teaches that Jesus was born to a virgin, as stated in Matthew and Luke, as if Stanley somehow disbelieves it or does not care if his audience believes it. If Mohler had listened to the whole sermon, he would know that the point of the sermon was to put the virgin birth in the context of God fulfilling His astounding promises throughout history, all the way to a virgin named Mary.
Mohler then adds accurate commentary of popular and academic attacks on the virgin birth of Christ in the context of Stanley's misquoted and out-of-context comments in order to associate him with liberalism.