Yes, it is as close to a fact as you can possibly get when talking about events occurring around 13.7 billion years ago. I guess it's still called a theory because we were not there. However, just like everything else, research it yourself and you'll find this to be true on your own. For those who don't want to or say they will and won't, I'll give you a quick break down in layman's terms (because I don't know all of the terminology myself, I just know how we know...)
Our scientists know the universe is expanding. Not only that, the expansion is accelerating. We were able to measure the distances between stars probably every year or so back then. Einstein's equations of general relativity, which led to the discovery that light bends, and the Hubble space telescope's first pictures of Doppler shifts in spiral galaxies, both helped contribute to the proving that distances were/are expanding between solar systems & galaxies. It has been almost 100 years now! We can probably measure these changes by the hour now. It is a FACT that our universe is expanding.
From here it is easy to see how we KNOW that the universe gets smaller as we go back in time. Complicated quantum physics and mathematics were able to reverse the accelerating expansion numbers to determine how long ago it was until the mass of the universe was the size of our pinkie nail. In order for the entire mass of our current universe condensed to that small of size, to end up at this massive sized universe, and along with the numbers from how long ago it had to be, we know there had to be a massive explosion. There is obviously more to it, but go wikipedia it if you're that far behind. What happened before the big bang is all that is in question now.
So, the Big Bang Theory was first proposed in the early 1930's by Lemaitre. It was still a theory back then. The Hubble space telescope and very qualified scientists/physicists like Albert Einstein, Alexander Freidmann, & Georges Lemaitre confirmed it's validity over the next couple decades. It is almost 100 years later. We know so much more. Scientists are far past trying to determine if the Big Bang FACT happened. It's time for everybody to catch up. The questions will only get deeper and more meaningful.
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