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The Big Bang model holds that everything that now comprises the universe was once concentrated in a single point of near-infinite density. Before this singularity exploded and the universe began, there was absolutely nothing - indeed, it's not clear whether one can even use the term "before" in reference to a pre-Big-Bang cosmos, as time itself may not have existed yet. In the current model, the universe began with the Big Bang, underwent cosmic inflation for a fraction of a second, then settled into the much more gradual expansion that is still going on, and likely will end with the universe as an infinitely expanded, featureless cosmos.

I'd love to have someone try to explain the mechanism behind this expansion.

If black holes, which are only part of this universe, exert such a strong gravitational field that even light can't escape, how do they explain how the entire mass of the whole universe could be contained within this minute point, and not be trapped forever in it by its own gravitational pull.

How did it escape itself? What kind of and amount of force was necessary to overcome that kind of gravitational attraction?

I think you might enjoy Russell Humphreys book:

Starlight and Time

I found it in my local library (quick read, small paperback) in the juvenile section. Tried to see if they might re-locate it but no dice since the author wrote it specifically for a younger audience. Probably assumed older generations would be too close-minded...

Re: Russell Humphreys
He’s a scientist who has made some startling and recently confirmed predictions related to yec.

Also excerpted from wiki...

Russell Humphreys, a young Earth creationist and a nuclear physicist, wrote a book called Starlight and Time, which attempts to explain the starlight problem with his ideas of how a young earth and universe can fit in with the distant starlight problem. The first starting point for Humphreys’ model is the original cosmic material, while the second concerns the state in which that matter was in, which Humphreys believes to have been a massive black hole. Humphreys argues at great length to the effect that the Big Bang theory does not and cannot begin with a black hole (due to the assumption of the cosmological principle). The model also suggests that the universe has a distinct physical “edge”, and that the Earth lies in the middle, something Humphreys believes is supported by claims of quantized redshifts.