Well, in the first place, we don't "know" that the Big Bang actually happened.* The notion that the Universe began in a single event which caused all of space-time and matter-energy to come into being is a theory. A theory is a proposed explanation for a set of observations. A theory can never be proven, but must be "testable" through observation or experimentation. Thus far, despite some notable problems, the Big Bang Theory has remained largely consistent with the observations and is widely accepted through the cosmological community.
The first observations leading to the Big Bang Theory involved objects we now call "galaxies," but which were, at the turn of the century, referred to as "spiral nebulae." American astronomer Vesto Slipher (1875-1969) observed that these nebulae were receding, or moving away from us. In 1912, Slipher ascertained this motion by studying the 'spectra' of these nebulae: the light they emit. The wavelengths of the light appeared elongated and so the lines within it were shifted toward the longer wavelength (lower energy) part of the spectrum.** Slipher's observations, alone, didn't precipitate the development of the Big Bang. We recall that in 1912, many astronomers believed that the Milky Way Galaxy defined the Universe's boundaries and that nothing existed beyond it.
Within the following decade, Albert Einstein introduced his General Theory of Relativity (1916), a comprehensive treatise on gravitation. The General Theory predicted that the Universe be static, a view espoused by Steady State Theorists who believed the Universe didn't have a specific beginning. They explained Slipher's recession observation as evidence that matter was created into the Universe at a specific rate and the recession was a response to its introduction.
In the 1920s, another American astronomer, Edwin Hubble (1889-1953), determined that the spiral nebulae, such as the famous one in Andromeda, were external galaxies. Improperly credited as having first observed galactic redshifts, Hubble nevertheless expanded the cosmic boundaries and, himself, determined that most galaxies receded from the Milky Way. He also established the Hubble Law, in which a galaxy's recession velocity increased with increasing distance.
Meanwhile, Russian mathematician Alexander Friedmann (1888-1925) derived the Friedmann Equations from Einstein's General Relativity and in so doing predicted that the Universe had to be expanding, in contrast to Einstein's static Universe mdel. In 1927, Belgian physicist-priest Georges Lemaitre (1894-1966) derived these equations himself. Citing the observations of Slipher and Hubble, Lemaitre theorized that the Universe was receding in response to an initial "explosion" event that created space-time, which had been regarded as separate until Einstein's Relativity theory conjoined them.
The notion that time had a beginning was appalling to most astronomers at the time. As a Roman Catholic Priest developed the theory, many objected to the religious aspect, believing Lemaitre was attempting to insinuate a Genesis event into astronomy. In the ensuing decades, two primary theories were espoused: the Steady State Theory and the Big Bang Theory. (Ironically, the preeminent Steady State Theorist, Fred Hoyle, coined the term "Big Bang," perhaps as a pejorative.) The Steady State insisted that the Universe didn't have a specific beginning, but that periodic matter creation could explain the recession. The Big Bang Theorists insisted that the cosmos took form and that the Universe expanded ever since.
They also cited other factors, such as the close agreement between the element abundances in the Universe and those predicted in "Big Bang Nucleosynthesis." This model predicts the abundance of the light elements, such as helium-4 and lithium-7 relative to hydrogen. Although not in as close accord with the lithium-7, the Big Bang Nucleosynthesis model accurate predicted the ratios of the other lighter elements.
In 1964, Arno Penzias and Robert Wilson inadvertently discovered the Cosmic Microwave Background Radiation, dubbed the "Big Bang Echo." This radiation, cooled to 2.7 K (2.7 degrees Kelvin above absolute zero), is believed to be a remnant of the initial radiant pervading the early Universe. The Steady State Theory didn't have a viable explanation, though it was wholly consistent with the Big Bang.
So, it is for these reasons that astronomers and cosmologists (astronomers who study the Universe entire) believe the Universe might have started in a Big Bang. We must remind everyone, of course, that other theories have been proposed, such as the M-Brane Theory, which states that the Big Bang occurred after the collision of hyperdimensional M-branes. Therefore, the Big Bang Theory might well be modified or dispensed with altogether, depending on what future generations of astronomers observe.
For now, it is the best explanation we have for how the cosmos came to be.
*Yes, we accept some of the blame for this issue. How often have we smugly said, "The Universe started in a Big Bang, approximately (fill in the blank) years ago." We have made that statement with inappropriate conviction. We can tell you with certainty that we humans occupy space-time (metaphysical philosophers need not reply), but we can't tell you with certainty how the cosmos began.
**Just a brief primer: a photon's energy is inversely proportional to its wavelength. The longer the wavelength, the lower the energy. If light is elongated through the Doppler Effect, it shifts toward the red part of the spectrum, which has a lower energy than the blue. Hence the term "red-shifted."