The big bang theory predicts the universe has cooled to a few degrees above absolute zero and is awash in a faint glow of the 3K microwave cosmic background radiation.
Why would the big bang cause the 3 degree Kelvin (3K) microwave cosmic background radiation? To answer that question we need to know something about the big bang theory. According to this theory, the universe originated about 15 billion years ago. At the instant the universe was created, the entire universe, all the space, energy, and matter, was compressed into a single point. The universe started to expand. It still expands.
In the beginning, the universe was very hot and dense. As it expanded it became less dense and cooled off, just as a hot gas will cool if allowed to expand.
In the early 1960s at the Institute for Advanced Study, in Princeton, New Jersey, Robert Dicke and his colleagues predicted that if the big bang theory for the origin of the universe were correct, the entire universe should have cooled to a few degrees above absolute zero (a few degrees Kelvin) and be awash in a faint microwave glow. Dicke's group happened to be closest in both time and space to Penzias and Wilson when they discovered the microwave cosmic background radiation, but they weren't the first to predict its existence.
In 1948 George Gamow, Ralph Alpher, and Robert Herman predicted from the big bang theory that the universe should have cooled to about 5 degrees above absolute zero (5 Kelvins), but didn't have the tools to search for the microwaves. Independently in 1964, Igor Novikov and Andre Doroshkevich of the Soviet Union suggested that the microwave remnants of the hot big bang should be detectable.
These groups studying the big bang from a theoretical standpoint, had predicted purely from the big bang theory that the universe should by now have cooled to just a few degrees above absolute zero. 3 Kelvin degrees above absolute zero is about 270 degrees below zero centigrade.
If an object radiates like an ideal blackbody, it will glow blue or white when at the temperature of a hot star . If it is at the temperature of a cool star it will glow red. At cooler temperatures it will still glow but at longer wavelengths. An object at room temperature glows in the infrared. At still cooler temperatures, say just a few degrees above absolute zero, the object will glow with radio waves at microwave wavelengths.
If the object is the universe, this faint microwave glow will permeate the entire universe. This microwave glow is what Penzias and Wilson discovered, quite accidentally in the early 1960s. The microwaves correspond to a temperature of 2.7 degrees above absolute zero, 2.7 Kelvins.
The radiation they discovered is now called the three degree Kelvin (3K) blackbody cosmic microwave background radiation, although most astronomers use one of several possible shortened forms of the full name. The fact that the theoretical prediction resulting from the big bang theory and the experimental discovery were made completely independently of each other increases our confidence that the big bang theory, or something very similar, is correct.
Hence, the microwave cosmic background radiation convinced most astronomers that the big bang theory was correct and that the steady state theory was incorrect. The theory already existed when Penzias and Wilson discovered the radiation. What then originally led to the first suggestion that the universe is expanding and originated in a big bang? No sensible person would suggest that the universe is expanding with out some good reason.
Barrow, J.H. and Silk, J., The Left Hand of Creation, Oxford, 1983.
Silk, J., The Big Bang, Times Books, 2000.