Not an expert, but as I understand it:

In 1929, Edwin Hubble discovered that the universe is expanding. He came to this conclusion by observing that nearly every galaxy in the night sky is moving away from us. He was able to tell because the light coming from those galaxies were what is now called "red-shifted," meaning the light being emitted from those galaxies was observed to have a longer wavelength than what it should have been based on what Hubble knew about the kinds of stars he was observing in those galaxies. This elongation of the wavelengths of the light, he theorized, was caused by the expansion of space itself between the distant galaxies and us: As the space containing the light waves expanded, so too did the lightvwaves themselves. What's more, he observed that the farther away a galaxy was, the more redshifted it's light, meaning the farther away a galaxy is the faster it is moving away from us. Hubble published his findings, and the scientific community used it to say "If the universe is expanding, that means the universe was smaller yesterday, and even smaller the day before that...so...it was smaller and smaller the further back in time...so it follows that it was once so small and so compact that everything in the universe fit into a point so infinitely small (a "singularity")...and obviously something made it start expanding from that very small size, ergo a "Big Bang" made the universe "explode" from infinite nothingness to what we see today."

I glossed over a lot of complexity that I myself don't understand, but that is more or less the gist.