1. The Expanding Universe
In the 1920s, Edwin Hubble saw that almost all galaxies were redshifted (moving away from us). The farther away they were, the more they were redshifted.
If galaxies are all moving away from us, yesterday they must have been closer, the day before even closer. If you take this to its logical conclusion, at some point in the distant past, the whole Universe must have been compressed into a very small volume, and it rapidly expanded.
This is where the idea for the Big Bang came from.

2. The theory is that the entire Universe was crammed into a tiny volume some 13 or 14 billion years ago. That volume “exploded”, or started a very rapid expansion. For the first several billion years, the Universe was too dense and hot for there to be anything but the simplest particles.
Eventually the Universe cooled and expanded enough for there to be huge clouds of matter, hydrogen. These clouds gathered together, gravitationally, and formed stars and galaxies. Today, there are about a quadrillion galaxies.

4. Recognizing that the farther away we look, the farther back in time we are seeing and also the more redshifted the light should be. If we look back 13.7 billion years ago, the beginning of time, light from the Big Bang should be redshifted all of the way into microwaves at 2.7 K. This was predicted in 1948 and discovered in the 1960s as the Cosmic Background Radiation, and it proves the Big Bang happened.

5. Two NASA missions have mapped the sky looking at the Cosmic Background Radiation, COBE and WMAP. They looked at the Universe just a few hundred thousand years after the Big Bang, and found structure.