After asking this question, I got confused. I always thought that repulsive gravity was a consequence of dark energy, the Nature of which is unknown. Many theories are flying around about its Nature. For example, that it's caused by a positive quantum vacuum energy who's density is constant and when the Universe gets big enough and the attractive gravity is diluted it will overtake the attraction of matter and make matter move away from each other, or by an emergent gravity, especially the one propagated by my fellow Dutchman Verlinde or in the inflation era by some dilaton field. The cause of this expansion (if it really exists, which is doubted by some physicists) might be unknown, but don't they all have in common that repulsive gravity is a consequence of an expansion of space?
I'm going to attempt a layperson's answer to this question, and it will bear little or no resemblance to Perelman's magnificent paper in Physics Letters B, to which I alluded in an earlier comment.
The simplest explanation of how gravity might become repulsive can be imagined through a reading of Poplawski's "Cosmology with torsion" (available on Arxiv) of 2010, which is based on Einstein-Cartan gravity, that provides a (tiny) spatial extent for fermions. (Although that theory relies on data that's not renormalizable, I believe its adaptations by Sciama and Kibble have left it consistent with reality.)
In the gravitational collapse (due to that complete expenditure of its nuclear fuel, which leaves it with pressure inadequate to balance its mass) of any large rotating star that's not unrealistically isolated, tidal effects will separate the particles of many virtual particle-antiparticle pairs between the opposite sides of the outward-propagating event horizon, with this separation materializing those on the EH's inboard side. As particles spin, their contact with the larger fermions of the star itself would accelerate some of them to the speed of light, which would not be consistent with their mass. Maupertius' Principle of Least Action would make an expansion of space infinitely more likely than any instantaneous addition of a comparable mass to all photons everywhere, so space expands. (This conformal change equates to a simple change in the viewpoint of all of its inhabitants everywhere, which affects them only upon the arrival of the resulting image within their particle horizons, and might, in our terms and through use of our current technology, be observationally indistinguishable from gravitational repulsion.)
Because the contraction of stellar collapse may balance the outward expansion that occurs after the spin-spin fermionic interaction that Poplawski describes mathematically, the Borde-Guth-Vilenkin Theorem, which requires that any universe (including "local" ones out of causal contact with any others) must have a beginning if it is "on average expanding", would not necessarily apply to our own. In a 2015 collaboration with Desai ("Non-parametric reconstruction of an inflaton potential"), that's also available for free on Arxiv, he demonstrates its consistency with the current CMB data. His cosmology is considered by Wikipedia to be a version or adjunct of inflation, although the inflaton particles would seem to be whatever remnants of the fermions of our LU's originating star may exist.