It comes from this realisation:
If you look into the sky, with good enough telescopes, you'll see that stuff is expanding -- stars and galaxies are moving away from each other.
But this is assuming that time is moving on too. So if you start moving backwards in time, you'll rewind the movement of the stars and galaxies, just like you'd see if you rewind a video.
So, you can imagine that if you keep rewinding, and everything is moving towards everything else, eventually you'll come to a point where everything is, and you can't rewind any more (the beginning of the tape, so to speak).
Next is time -- why do we think that this is the beginning of time? Well, it's all down to gravity. Everything that stuff is made of has some gravity. So if you have more stuff, you have more gravity. The Sun has more gravity than the Earth, because there's a lot more of it.
It was found (and is predicted quite accurately by Einstein's General Relativity), that a clock on the highest mountain runs a little faster than a clock at sea level. So, if you move away from a gravitational source, time runs faster. Kinda weird, but true.
But this also means that if you have a clock at somewhere with a lot of gravity, time will run really slow. So, at the time of the big bang, all the matter of the universe was there at the same point, so a clock at that point would have slowed enough to have stopped.