Red shift.
We've seen that the universe is expanding. That means, when played backwards, everything comes to a single point. This is the 'Big Bang'.
This left two possibilities: a finite expanding universe or a form of solid state in which ever more matter/energy came form somewhere like a white hole.
The law of the conservation of energy rules out the second possibility. Meaning the amount of matter/energy in the universe is finite.
Through mathematics that I'm not going to claim to fully comprehend, it has been determined that 'TBB' was not merely matter moving through space, but rather the expansion of space-time itself.
We're reaching the point where the mathematics involved became too much for me and I stopped trying to grasp all the details.
also, the expansion of space-time and the finite nature of matter/energy are what bring about the whole entropy thing