"In the beginning..."
There is much we don't know about the initial conditions in the first infinitesimal fraction of a second of the universe's existence, but there is much that we can know about the period immediately following. We know that early on the matter in the universe was compressed into such a small space that the resulting temperature would have prevented atoms from forming. Because electrons at that point would have existed as a formless cloud in a very nearly uniform distribution of free protons and neutrons, there would have been no "free path" for photons. Light, as we are accustomed to experiencing it, couldn't have existed. If an observer could have been present in this observer-less universe, they would have seen nothing more than a uniform grey fog.
That uniform fog in an atom-less universe persisted for hundreds of thousands of years (current best estimate is 380,000), and all the while the universe continued to expand. As the universe expanded, it cooled.
That 380,000 year number is an estimate, but what happened next is not. As that nearly featureless (the technical word is "isotropic") universe cooled, it eventually reached 2700 degrees K, which is the temperature at which electrons can bind to protons, creating hydrogen atoms. Since the entire universe was nearly uniform in temperature, that binding (misleadingly termed "recombination") occurred everywhere, all at once. When a high energy electron binds to a proton (and settles into a ground state), a quantum of the electromagnetic field, a photon, is emitted.
In that moment, the cosmos as we know it was born in a vast and universal flash of light.
It's a beautiful and poetic metaphor, isn't it? Everything, everywhere being born in a flash of light? The question is, how do we know it's true? How do we know it's not just a pretty story?
The description I just gave you implies some testable predictions. The most obvious of those predictions is that we should still observe the universe expanding. We do observe that of course. (A quick shout-out to Edwin Hubble for discovering it and giving us the first measurement of the rate of expansion, termed the "Hubble Constant"). We can't really use Hubble expansion as a test for our "Big Bang" hypothesis though because it was the observation of that expansion that lead us to hypothesize the Big Bang in the first place.
So what other predictions are implied by the Big Bang? (ok, technically, the "Lambda CDM model").
A very quick detour to explain black-body radiance: light can interact with matter in a limited number of ways. A photon, when encountering an atom, can be absorbed (and possibly re-emitted later) or reflected... and that's it, those are the options. Imagine an object that doesn't reflect light at any wavelength. Ignoring for a moment any light that object emits, that object would be perfectly black. A perfect "black-body" is an object that can emit photons, but cannot reflect them. Black-bodies are sometimes called "perfect emitters" and stars, by the way, are near perfect black-bodies. (Yeah, I know... English is weird).
Everything in the universe above the temperature of 0 degrees K (and that is, well, everything in the universe) emits light. (Here I'm using "light" to include photons of any electromagnetic wavelength). The particular pattern of photon wavelengths that are emitted is called "black-body radiance". For a given temperature, not every wavelength is emitted at the same intensity, and there is a well defined shape of a black-body "wavelength vs. intensity" curve, and there is always a well defined peak. The wavelength of the peak in a black-body radiance curve depends on one thing and one thing only: the temperature of the object doing the radiating. If you know the temperature of the object, you can predict with extraordinary accuracy what the peak wavelength of that curve is. If you know the wavelength of the peak in that curve, you can predict with extraordinary accuracy the temperature of the object emitting it. Peak wavelength --> temperature, and vice versa. (For an everyday demonstration of this, watch a video of steel being heated. One can tell the temperature of the steel by examining its color, and one can predict the color steel will be at any given temperature)
So... for black-body photon-emission: temperature => wavelength, and wavelength => temperature. Is everyone still with me? Good!
So what other predictions are implied by the big bang?
The most obvious one is that since the universe was a (very nearly) uniform temperature when photon-decoupling (i.e., "the universe-sized flash-bulb") occurred, the spectral radiance of that light should exhibit a nearly perfect black-body radiance curve. And that light should still permeate the universe in all directions, everywhere.
Another prediction is that since we have a good estimate of the rate of expansion of the universe and we have a good estimate of how long it's been expanding, we should be able to predict how far the wavelengths of those photons have been stretched out. (As space expands, the wavelength of the photons expands with it, like two children pulling on opposite ends of a slinky). For the record, and without doing the math, if that "flash" really happened, those photons should now have a wavelength of about 2mm. (2mm is in the E-M wavelength range we usually call "microwaves", and that wavelength corresponds to a black-body temperature of right around 3 degrees K)
Another prediction is that because that primordial soup of free electrons, protons, and neutrons just before photon decoupling (i.e., "the big flash") was almost but not quite uniform (due to a quantum effect called "vacuum fluctuations"), we should still be able to see those very (very) small wavelength/temperature variations when we look for this "cosmic microwave background" radiation. (In much the same was that complete uniformity is called "isotropy", these small deviations from uniformity are called "anisotropies")
Another prediction is that variations in the density of matter we see in the universe around us (observed as the distribution of galaxies in the observable universe) should correspond to the very small variations we measure in the cosmic microwave background.
And last but not least... since early galaxies formed at roughly the same time and the light from very distant galaxies takes longer to reach us than the light from nearby galaxies, from our perspective, very distant galaxies should have a higher density of young, blue stars than nearer galaxies do. (Note that we have to calibrate-out expansion-induced red-shift when we do this comparison)
Every one of those predictions (as well as others that I haven't mentioned) has been tested, and every one of those predicted observations has been observed with very high degrees of precision.
This testable, tested model that gives us a real understanding of origins and evolution of the physical universe is one of the greatest achievements of the human mind, and stands as a testament to what human beings can discover.