Not really sure I understand bookbinder's point. Suppose for the sake of argument you came across a whole bunch of rubble. You find bricks and mortar dust; material that could have been a carpet; various traces of wood; metal, some of it that was evidently once part of some piping. And so on, and so forth. At some point, staring at all this wreckage, you might eventually conclude that a house once stood here, but that it had blown up or been destroyed somehow. With a bit of work, you might even be able to make some decent estimates of how large the house was, and what sort of explosion might have been responsible for the damage seen. One way of doing this might be to construct your own miniature houses and blow them up, to see what gets leftover, and in doing so you can get a better idea of the shape of the house, its internal dimensions, how much wood and metal was inside, and whereabouts it was, etc etc. You can do all this and you don't need anyone to see that the house actually blew up.
In many ways, the same sort of thing is true of the Big Bang. The birth of the Universe is such a colossal event that it naturally can be expected to leave traces. Among other things, those traces include the CMB radiation. Although this was in fact formed a few hundred thousand years after the Big Bang, it nevertheless can be used to make some observations about the Big Bang (and other theories about the Early Universe) as each competing theory can be expected to leave distinct traces in that CMB.
Additionally, there is the point that the Universe is still expanding, which to some extent might be because the explosion hasn't finished yet (rather like, while standing by the rubble, a stray brick randomly falls out of the sky at your feet). And then to test some of the assumptions, various particle physics experiments essentially consist in smashing particles together at stupidly high energies and looking at the stuff that flies off. This is, hopefully, creating the conditions of the Early Universe a very short time after the Big Bang, and again allows you to test the competing models.