Much of the evidence is beyond the technical scope of this BLOG. There are more than a dozen different things that support the Big Bang hypothesis. Some of the evidence includes the Sunyaev-Zel'dovich effect and the Integrated Sachs-Wolfe effect. Understanding those things is as daunting as their names imply.
But some of the pieces of evidence aren’t so difficult to understand.
One hypothesis, not too difficult to believe, is that at the moment of the Big Bang the universe was very, very hot. As it expanded, it cooled off.
But, like a campfire, it should have never cooled off to nothing. This is a testable hypothesis. If you look around the universe, you should see some sort of residual heat. It should be very small, but measurable nonetheless. It should also be very, though not necessarily perfectly uniform. (It should also have a black-body spectrum.)
That is exactly what we see.
This was predicted ahead of time but was originally discovered, quite by accident. Here’s a description of what happened:
"In 1964, Arno Penzias and Robert Wilson were experimenting with a supersensitive, 20 foot (6 m) horn antenna originally built to detect radio waves bounced off echo balloon satellites. To measure these faint radio waves, they had to eliminate all recognizable interference from their receiver. They removed the effects of radar and radio broadcasting, and suppressed interference from the heat in the receiver itself by cooling it with liquid helium to −269 °C, only 4 °C above absolute zero.
"When Penzias and Wilson reduced their data they found a low, steady, mysterious noise that persisted in their receiver. This residual noise was 100 times more intense than they had expected, was evenly spread over the sky, and was present day and night. They were certain that the radiation they detected on a wavelength of 7.35 centimeters did not come from the Earth, the Sun, or our galaxy. After thoroughly checking their equipment, removing some pigeons nesting in the antenna and cleaning out the accumulated droppings, the noise remained. Both concluded that this noise was coming from outside our own galaxy--although they were not aware of any radio source that would account for it.
"At that same time, Robert H. Dicke, Jim Peebles, and David Wilkinson, astrophysicists at Princeton University just 40 miles (60 km) away, were preparing to search for microwave radiation in this region of the spectrum. Dicke and his colleagues reasoned that the Big Bang must have scattered not only the matter that condensed into galaxies but also must have released a tremendous blast of radiation. With the proper instrumentation, this radiation should be detectable.
"When a friend (Bernard F. Burke, Prof. of Physics at MIT) told Penzias about a preprint paper he had seen by Jim Peebles on the possibility of finding radiation left over from an explosion that filled the universe at the beginning of its existence, Penzias and Wilson began to realize the significance of their discovery. The characteristics of the radiation detected by Penzias and Wilson fit exactly the radiation predicted by Robert H. Dicke and his colleagues at Princeton University. Penzias called Dicke at Princeton, who immediately sent him a copy of the still-unpublished Peebles paper. Penzias read the paper and called Dicke again and invited him to Bell Labs to look at the Horn Antenna and listen to the background noise. Robert Dicke, P. J. E. Peebles, P. G. Roll and D. T. Wilkinson interpreted this radiation as a signature of the Big Bang.
"Later a satellite, called the COsmic Background Explorer (COBE), was launched. It made very precise measurements for years in space that matched the expectations of the cosmologists."
Another piece of evidence that isn’t too difficult to understand is the presence of varying amounts of very light elements.
One hypothesis is that soon after the Big Bang a higher proportion of lighter elements were present. Heavier elements (dare I say it?) evolved from these lighter elements over time.
We can effectively look back in time. Since light moves, of course, at the speed-of-light, if we look at a star that is 100 light-years from Earth, we are seeing things as they were 100 years ago. If we look at a star that is a billion light-years from Earth, we are just now viewing that star as it looked a billion years ago.
It is possible to identify specific elements when looking at light because each has a specific spectral pattern.
Therefore, we can look at the light from stars at varying distances from Earth and measure the percentage of lighter elements present on those stars.
The percentages of helium and hydrogen are somewhat suspect, because stars are fueled by fusion energy that involves the conversion of hydrogen into helium.
But lithium, the next lightest element, has no such problem. We see higher percentages of lithium in stars created closer to the time of the Big Bang, exactly as predicted by the theory.
As I say, there is quite a bit more evidence supporting the Big Bang. Very few scientists who have objectively looked at it all doubt that the Big Bang took place.
 http://encyclopedia.thefreedictionary.com/Discovery%20of%20cosmic%20microwave%20background%20radiation, referenced on January 15, 2009