I'm not going to call BS on this right away...
...but could someone please clarify this for me because it makes absolutely no sense. How is it that a black hole, which is a collapsed star, could be responsible for the formation the first generation of stars in the universe? Isn't the very existence of a black hole predicated on there being stars in the universe?
A black hole doesn't have to be formed from a collapsed star. A black hole is any gravity well so deep that matter and electromagnetic radiation cannot escape from it*.
Creation of primordial black holes in the big bang
Gravitational collapse requires great densities. In the current epoch of the universe these high densities are only found in stars, but in the early universe shortly after the big bang densities were much greater, possibly allowing for the creation of black holes. The high density alone is not enough to allow the formation of black holes since a uniform mass distribution will not allow the mass to bunch up. In order for primordial black holes to form in such a dense medium, there must be initial density perturbations which can then grow under their own gravity. Different models for the early universe vary widely in their predictions of the size of these perturbations. Various models predict the creation of black holes, ranging from a Planck mass to hundreds of thousands of solar masses. Primordial black holes could thus account for the creation of any type of black hole.
Source
Once these primordial black holes formed, so goes the theory, their high gravity attracted matter. Not all of it would fall into the hole: some would simply fall into orbit around it. After a while, further bunching and clumping would cause this matter to coalesce into stars.
That's one theory of primordial star formation; there are others that don't involve black holes.
*Not strictly true because of the so-called Hawking radiation, but that needn't concern us here.
