Actually Einstein predicted the Big Bang before Hubble found the first observational evidence.
Einstein found that in his theory of General Relativity that the universe as a whole should be either expanding or contracting. He didn't like this so he added an extra term the "cosmological constant" to balance the equations and have a universe that was not expanding or contracting. That sort of static universe was what was expected in those days.
Meanwhile, Edmund Hubble began making observations in the 1920s from the Mt Wilson Observatory of distant "nebulae." He had discovered previously that many of this faint blobs, previously thought to be gas clouds, were in fact galaxies like our own Milky Way. This was a big surprise since previosuly the Milky Way was the only galaxy, or what they called at the time, an "island universe."
So, it was natural for Hubble to look at the spectra of some of these distant galaxies. What he found was that, except for a couple of very nearby galaxies like M-31 the Andromeda Galaxy, all the other galaxies had shifted spectra. That is, these other galaxies has spectra which were redshifted, indicating that they were moving away from us.
Further he found that the fainter galaxies (which on average are farther away) had the largest redshifts. This lead to the famous "Hubble's Law" that says that the observed redshift is proportional to the distance a galaxy is from us.
Einstein found out about Hubble's discovery and realized what it meant - the whole universe is expanding. Although naively it looks like everything is moving away from us, all of space is expanding, so that an observor in any galaxy would detect the same thing - that every galaxy is moving away from them.
The usual analogy is to think of rasins in rasin bread. As the bread rises all the dough (space) expands, so the distance between all rasins (galaxies) grows. Thus from any one rasin (galaxy) all the rest look like they are moving away.
Einstein realized that his "cosmological constant" was unnecessary. He famously called it his "greatest blunder" and that he could have officially predicted the expansion of the universe before Hubble confirmed it with observations. (And in fact, before Hubble's discovery, other scientists like Willem de Sitter and Alexander Friedmann used the equations of General Relativity to calculate an expanding universe.)
At that point it was realized that if space is expanding that in the past it must have been all bunched up in one spot. Just before Hubble Georges Lemaître realized this and thought that at first there was some sort of "primeval atom" from which everything else came from.
Fred Hoyle didn't like Lemaitre's idea at all, not the least of which was because Lemaitre insisted that God started the whole thing. Hoyle developed a rival theory the Steady State universe, and dismissed the other type of ideas as a "hot big bang." Of course Hoyle's term, meant as a put-down, became the offical name. (Hoyle's Steady State Theory was a serious rival to the Big Bang until the 1960s and the discovery of the microwave background.)
It wasn't until 1948 that the modern aspects of the Big Bang were proposed by George Gamow and Ralph Alpher. These guys showed that from an initial explosion of compressed everything one would get the correct mixture of hydrogen and helium and other trace elements observed in the whole universe. That is, if one starts from a hot subatomic, compressed mixture and lets it go, it will recombine into the right ratio of atoms. (Right being the ration actually observed.)
And so it goes from there, to much more observational evidence for the Big Bang, as well as lots of theoretical progress. At the moment we know that the old cyclical idea, that the universe expands and then contracts, is wrong. In fact the universe seems to be accelerating.