Because cosmology is the study of the origin and evolution of the entire universe, there are some aspects that we can not actually test. We must assume that they are correct.
Three of these assumptions comprise the cosmological principle. They are:
In a nutshell, the cosmological principle means that the universe looks essentially the same at any location in the universe, but the universe can change with time.
The perfect cosmological principle takes these three assumptions and adds the fourth assumption that the universe does not change with time. It looks essentially the same for infinite times in the past and future as it does now.
The cosmological principle is the foundation for the big bang theory, and the steady state theory rests on the perfect cosmological principle. So the universe can evolve or change in the big bang theory but not the steady state theory.
There are some fundamental astronomical observations that are relevant to cosmology. In some cases they can support or refute one of the theories.
In the 1920s, Edwin Hubble noticed that more distant galaxies are moving away from us at a greater speed. His observation tells us that the universe is expanding. A graph with the distance to the galaxies on the horizontal axis and their recessional speeds on the vertical axis is the Hubble plot. The slope of this plot, the Hubble constant, tells us how fast the universe is expanding.
Both the big bang and the steady state theories are consistent with Hubble's observation. Hence the fact that the universe is expanding can not be used against either theory.
The Hubble plot does however provide some evidence that the universe changes with time. For the most distant galaxies, the graph seems to curve upward a little. This indicates that the expansion rate was slightly faster in the distant past than now. These observations are however very difficult.
Quasars are the very energetic nuclei of distant galaxies and are among the most distant objects in the universe. We do not observe quasars less than about a billion light years away and most are farther. Because of the time it takes their light to reach us, the lack of nearby quasars means they no longer exist.
These examples of evolution in the universe are evidence against the steady state theory, which is based on the perfect cosmological principle.
Arno Penzias and Robert Wilson, working for Bell Labs in the 1960s, designed a low noise microwave receiver. Despite all attempts there was one source of noise (static) that they could not eliminate. They even climbed into the antenna to remove some nesting pigeons and the "white dielectric film" they had deposited on the antenna.
This microwave noise had properties of a blackbody radiator (basically something emitting heat radiation) at a temperature of 3 degrees above absolute zero. Independent theoretical calculations predicted that if the big bang theory were correct, the universe should have cooled to a temperature of a few degrees above absolute zero.
Rather than being pigeon droppings, the source of the noise Penzias and Wilson observed was the left over heat radiation from the big bang. They accidentally discovered the universe.
Because this microwave background radiation had been independently predicted by the big bang theory and could not be easily explained by the steady state theory most astronomers were convinced that some version of the big bang theory is correct. The background radiation dealt the death blow to the steady state theory.
Further Reading
Harrison, Edward, Cosmology The Science of the Universe, Cambridge University Press, 1981
Zeilik, Michael, Astronomy the Evolving Universe, 9th edition, Cambridge, 2002