The first: The expansion of the universe. The universe is expanding now, so in the past, it must have been smaller. If it were smaller in the past, then there probably was a time when it was infinitesimally small. One could ask why don't we think that it might be expanding now but it could have been shrinking before and we just don't know about it. The answer is that there is simply no mechanism that we know about that could accomplish this transition on a universal scale.
The second line of evidence is the Cosmic Microwave Background Radiation (CMB) that was discovered in 1965 by Arno Penzias and Robert Wilson from Bell Labs. They were working with a microwave receiver but were getting noise from every direction they pointed the receiver. It was coming from all over the sky at what seemed to be exactly the same frequency. This was the first evidence for the CMB, and they later shared a Nobel Prize for this discovery.
The third major pillar of the Big Bang theory lies in the abundance of the different elements of the universe. The theory predicts that certain amounts of hydrogen, helium, and other elements should be made. Observations have shown almost exactly the amounts that are predicted.
The fourth piece is that the Big Bang theory is the only theory that comprehensively lays down a framework for the eventual evolution of the universe as we observe it today.
