What evidence supports the big bang theory?

There are several aspects of the Big Bang Theory supported by observed phenomena:

• The current universe is presumed to have a beginning state.
• Some galaxies appear to be moving away from us at speeds proportional to their distance. This is called "Hubble's Law," named after Edwin Hubble (1889-1953) who discovered this phenomenon in 1929. This observation supports the expansion of the universe and suggests that the universe was once compacted.
• The initial, very hot state that followed the Big Bang should have left some remnant of its heat. In 1965, Radioastronomers Arno Penzias and Robert Wilson discovered a 2.725 degree Kelvin (-454.765 degree Fahrenheit, -270.425 degree Celsius) Cosmic Microwave Background radiation (CMB) which pervades the observable universe. This is thought to be the remnant which scientists were looking for. Penzias and Wilson shared in the 1978 Nobel Prize for Physics for their discovery.
• The abundance of the "light elements" hydrogen and helium found in the observable universe (which would have been the first elements formed) is thought to support the Big Bang model of origins.

The idea is that, since galaxies are moving away from each other now, then they used to be closer to each other. Extrapolating backwards in time, one can go back far enough to a time and place where all matter resided in a very small, dense region. called a singularity.
This state would have been all energy, as matter could not have existed as we know it in an energy field so intense. The "Big Bang" was the release of that energy, creating the current era of spacetime.
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.

Replacing The Steady State Theory

The Steady State Theory was an early, only partly scientific proposition put forth to explain how the universe around us created matter on a continual basis. Founded in large part on the ancient idea that our universe was a firmament, and only subject to margional changes, like those caused by us, the explosion of a star, or the effects of gravity as it pulls back the matter and energy from those stars and creates new objects, the Steady State Theory basically told us that the universe was always here and always would be...and also would remain the same size.
The later advancement of science in new areas such as quantum theory, relativity, and Edwin Hubble's discovery that the universe was actually expanding, all pointed in a direction different from that of the Steady State Theory. This led astronomers and cosmologists to begin pondering what the universe would have looked like before its expansion. At some point in the past the universe must have been much smaller, but how much smaller could it have been?
Quantum mechanics allows us to imagine and make practical predictions about how tiny the universe could have been in the past, which greatly helped the Big Bang Theory to catch on. It was suggested that the universe was at one point so tiny that it was the size of the tip of a needle, and was referred to as a singularity, much like the singularity of a black hole. But a black hole limitlessly pulls in everything around it, and there's no evidence of black holes exploding and creating new universes within our own. So the singularity idea faded away, though almost painfully slowly in some circles.
Journalists, and sometimes scientists, will erronously state that there was once nothing, and then the Big Bang created the universe. This isn't correct. There was never "nothing." You can look at this in two ways. 1) Energy can be created from matter, and matter can be created from energy. In fact, matter pushed to the speed of light literally becomes energy. Thus, even if the universe were crushed under all it's own gravity at the speed of light into energy 13.7 or so billion years ago at the place where the big bang occured, there was still energy. There was still something. Then there's option 2) Maybe God created the uinverse, but He would still have been here before us so there could never have been nothing.
With option 1, the universe began as energy packed so tight that even our theories of quantum mechanics cannot agree on what might have happened. It seems logical, at least on our common macroscopic level, that energy packed so tight would explode and begin expanding outward. And if we started from energy, that energy had to come from somewhere. Perhaps from some prior universe.

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