In short, the universe is expanding. All of the galaxies we have observed that are farthest away are moving away from us. And the farther away they are, the faster they are moving away. It took little imagination to "rewind" that picture and see a time when all of the matter and energy of the universe was basically "in one place" and there was no universe. With all the energy (matter could not have existed as we know it in an energy field so intense) in that one place after arriving or being created, a huge explosion, the Big Bang, created space-time.

The energy dispersed and, as things cooled down, matter began to form. It's the expanding universe that is the evidence for the Big Bang. That and the microwave background radiation that was discovered in the 1960's (and found to be everywhere) that is convincing. Here's an analogy (though not necessarily the best one) for the microwave background radiation. Imagine a cannon shell exploding in the air. After a few seconds, the blast is "all gone" but there is a cloud of smoke everywhere. The microwave background radiation is like the "leftover smoke" of the high (very, very, very high) energy event that was the blast of the Big Bang.