Unfortunately the article seems to gloss over that point with just one sentence:

The conclusion is profound: Croker and Weiner had already shown that if k is 3, then all black holes in the universe collectively contribute a nearly constant dark energy density, just like measurements of dark energy suggest.

That sentence contains a link to a paper[1] that's highly technical and beyond my understanding of physics, but concludes:

A population of such stellar-collapse remnants can shift in energy $\propto a^3$ while diluting in number density $\propto 1/{a}^{3}$. The population-averaged energy density is then effectively constant and readily produces the observed ${{\rm{\Omega }}}_{{\rm{\Lambda }}}$.

which matches the claim in the article.

But to understand why, I guess we'd have to learn general relativity to understand the math in that paper.

1: https://iopscience.iop.org/article/10.3847/1538-4357/ab32da#...