One of the basic tenets of science is that there is only a finite amount of matter & energy within the universe. The universe is essentially a closed system, and that whatever matter & energy was present in the big bang explosion is all there ever will be for it to use. But how do scientists know this? Is this theoretical (ergo, scientists THINK this is accurate), or factual (they KNOW it to be true)? And when was this tenet first hypothesized/discovered/assumed/whatever?
Every one of these assertions requires lots of caveats. For example, there are some scientists who have demonstrated how an infinite universe could have resulted from the Big Bang, although I don’t understand any of it well enough to discuss it. They use more advanced and inclusive mathematical constructs than relativity.
Relativity theory, however, says two things. One is that the speed of light is the absolute upper bound for anything made of mass, and the other is that mass and energy are one and the same.
That means that although the fabric of space can move faster than the speed of light, as it appears to have done during the period of inflation, the subsequent expansion of the universe is limited by the amount of time that has elapsed since the Big Bang. Other aspects of relativity theory limit the possible structure of our real world universe to a few possibilities and I believe all are finite. The universe may also be bounded, but that’s another discussion.
The equivalence of mass and energy gives rise to the Law of Conservation of Mass-Energy, which states that mass-energy can not be destroyed or created. That’s why the total amount has not changed. Conservation laws are based on gauge symmetries at which point I’m in way over my head. You can also look at the universe as information, as quantum mechanics does. QM also says that information cannot be destroyed, and the latest theories of black holes appears to confirm that not even a black hole can remove information from the universe.
Whether string theory or loop quantum gravity or the other mathematical constructs that attempt to combine relativity and QM into a larger package contradict any of this is for the physics pholk to say. I just wanted to set down a few words in English before the equations start flying. 
It is a basic tenet? I thought this was not definitively decided.
It is clear that the observable universe is finite. It can be no bigger to an observer than light could have travelled since the Big Bang. If the Universe is ~13.5 billion years old than the Universe has a radius of 13.5 billion light years (actually I think it is more due to the universe expanding but someone who knows better will have to say). Regardless of the size anything outside of the sphere of the Universe centered on your head is meaningless as it can have no effect on us…in effect it is as if nothing is there even if there is.
So, as a practical matter the Universe is finite. The reality may be different but not sure if we can ever know the truth of it.
This is not a basic tenet of science. There are some cosmological models which require the total energy of the Universe to be finite, but these models are not well-established, and there are other models which are perfectly consistent with an infinite Universe. In fact, current observations seem to favor an infinite Universe, though that’s not (and cannot be) conclusive.
The law of conservation of energy is not stated globally, and in fact, for some cosmological models, cannot be extended globally. All modern descriptions of conservation of energy are local. Roughly, if one defines some small region, and the energy within that region increases, then energy must have come in, on net, through the boundaries of the region (likewise for energy decreasing, of course).
In fact, all of the laws of physics are defined locally, which is why it’s not a problem that some regions of space are receding from us at greater than the speed of light (this is presumably still happening today, incidentally, not just at the time of inflation). One still cannot define a small region containing two objects with a relative speed between them of greater than c.
What’s done is to show that the divergence of the mass-energy field is zero. This means that in any four-dimensional box the amount of energy leaving at the future side minus the amount entering on the past side equals the amount of energy that came in through the other sides. In other words: that energy is conserved. This (purely local) result has been verified to an amazing degree of precision and it would be simply astounding if it didn’t hold throughout spacetime. In particular, all the way back to the origins.
For observers following the scientific method to “know” anything it has to be repeatable- i.e., it has to be observed multiple times under constant conditions.
Clearly no astronomer or physicist could have done this with the universal mass.
This is a classic example of the misconceptions engendered by the standard public school treatment of “the scientific method”. What a scientific theory must do is make repeatable testable predictions. If we took your objection seriously, nothing in cosmology could be “known”.