I missed the program but I don't think the Big Bang is losing any momentum with anybody mainstream - the evidence for it is way too strong.
the point often misunderstood is that time is bound up with space so although if you extrapolate back the expanion of the Universe back you get to the "Big Bang" if you were in the Universe running the clock back time would go slower and slower until it stopped at that point - so the Universe in a very real way has always been here.
Issues with time are very hard to get to grips with.
As for what the Universe is expanding into try this as an exercise
You are a two dimensional triangle living on the surface of a hot air baloon. You are a small triangle so the baloon looks nice and flat to you.
But the baloon continues to inflate - your Universe gets larger and larger and you can't work out where all this extra space keeps coming from.
This is similar to one idea that the Universe may be a 4D torus ( ring doughnut ) shape.
This would mean that if you could go far enough in any direction you'd come back to the same point.
here's a video of a 4D torus