How did scientists come up with the Theory of the Big Bang?
Gravitation is the most dominant force that shapes the Universe and decides how it evolves. Newton's model of Gravitation would not allow the construction of Cosmological Models and was plagued by instabilities. Einstein's model of Gravitation called The General Theory of Relativity allowed the construction of self consistent Cosmological Models for the first time.
But General Relativity based Cosmological Models always came up with dynamic Universe (either expanding or contracting). Einstein had to introduce a Cosmological Constant artificially to make the Universe static. But the natural prediction of General Relativity was a dynamic Universe that is either expanding or contracting.
On the observational side in 1920s Edwin Hubble showed that all galaxies (except a few that were too close to milkyway) showed redshift and the farther a galaxy is from us higher is its redshift. This meant that galaxies are drifting away from us and the farther a galaxy is from us higher is its velocity of drift. The simplest interpretation of this observation is that of an expanding Universe.
In late 1940s George Gamov reasoned that if the Universe is expanding today it would have been denser at a earlier time and so must have been hot. This is the idea that is called the Hot Big Bang Theory.
In 1948 Ralph Alpher and Robert Hermann predicted that when the hot Universe cooled with time because of expansion at some point the temperature would have reached a point where electrons can become bound to nuclei and form neutral atoms. At that time the Universe would become transparent to light and the thermal radiation from that hot phase would have been redshifted today to microwave wavelengths and should be measurable as a Cosmic Microwave Background Radiation with a Planck Spectrum. They estimated its present day blackbody temperature to be around 5K.
This radiation was discovered in 1964 by Arno Penzias and Robert Wilson who found its blackbody temperature to be near 3K.