He doesn't need to explain it, because Borde-Guth-Vilenkin themselves not only don't claim any such thing, they actually wrote:
"What can lie beyond the boundary? Several possibilities have been discussed, one being that the boundary of the inflating region corresponds to the beginning of the Universe in a quantum nucleation event."
Vilenkin himself suggested one such possibility. http://mukto-mona.net/science/physics/a_vilinkin/universe_from_nothing.pdf
There are others. There is the possibility, for example, that time is finite but has no boundary. Another possibility is that time has only recently become time-like (in the sense of Lorentz invariance) and at the boundary was actually a space-like dimension. The authors you cite actually don't talk in the absolute terms you're suggesting; and they certainly don't rule out a uniquely quantum beginning to the universe.
He does not need to......Well, I suppose you are right, but it does not change what the theory says. What you quoted was a paper Vilenkin published (not Borde/Guth) in 1982. In the October 1, 2001, they published "Inflation Is Not Past-Eternal" . http://arXiv:gr-qc/0110012v1.
A watershed, of a sort, came with this publication where Borde, Guth, Vilenkin formulated their theory estabishing that any universewhich has on average over its past history been in a state of cosmic expansion. Theorists intent on avoiding an absolute beginning of the universe could previously take refuge in the period of time known as Planck time, an era so poorly understood that one commentator has compared it with the regions of the maps of ancient cartographers marked "Here there be dragons!" But the Borde-Guth-Vilenkin theorem does not depend upon any particular physical description of the universe prior to Planck time, being based on a deceptively simple physical reasoning which will hold regardless of our uncertainty concerning that era. It sweeps away the most important attempts to avoid an absolute beginning of the universe. Vilenkin pulls no punches. "It is said that an argument is what convinces reasonable men and proof is what it takes to convince even an unreasonable man. Moreover, the Borde-Guth-Vilenkin theorem requires that the multiverse itself cannot be extended into the infinite past.
I guess Vilenkin learned a lot in 20 years, or from his collaboration with Borde and Guth. I addressed a quantum beginning earlier.