This is actually a common question. Many websites have been setup to try to explain this. I like this one for instance. I shall attempt to do my own layman explanation.
First of all, in order to have a black hole, you need to have a place for it to be in. Since there was no such thing as a universe, there isn't a place for the black hole to actually exist in. It's like asking, what is north of the north pole. There is no reference for an answer.
Secondly, gravity and other fundamental forces didn't act the way we are used to them acting. All four fundamental forces were combined in one basic force. Therefore there was no such thing as gravity to actually act on the mass as it existed. And then there is the problem that there was no actual mass. It was energy, which was creating the density of the universe. I know it's kind of counter-intuitive to how we are used to understanding these terms. Actually, Brian Greene has a good explanation of all this in his book Elegant Universe. Suffice it to say, at the start of the universe, the fundamental forces really acted very differently from how we see them now. As gravity was separating itself from the other forces, it actually had a repulsive effect as opposed to attraction.
Finally, the big bang wasn't really an explosion or a bang as one would think of it. It's actually the rapid expansion of space itself. Instead of things themselves speeding away from each other in a fixed space, the motion is actually caused by the space between the objects themselves getting bigger. Visualizing this is usually presented as a balloon expanding, and seeing how two dots on the surface of the balloon get further away from each other without actually moving on the surface. Of course, it's harder to visualize this in three dimensions (or actually four). While we are limited to the speed of light for any objects, the expansion of space itself is not limited by this (Brian Greene's book also has info on this).
The bottom line is that anything that you think you know should probably be discarded when thinking about the big bang. The reason that there is "string theory" and many other things is that classical quantum physics and relativity physics break down at the Plank Epoch which is at the heart of the big bang.
I hope that helps.