Couldn't the existence of a multiverse explain the beginning of the universe?

Yes, a multiverse of some kind might explain why our universe is here. That's possible. But really, that just pushes the question back further. How did the multiverse begin then? As noted in the video, the Borde-Guth-Vilenkin theorem proved that any expanding multiverse must have had a beginning as well. So we end up just replacing the "universe" with the "multiverse" in the argument and realize that the multiverse points us to God as well.

To escape the beginning of the multiverse, some people have proposed a sort of "multiverse machine" that has been creating new universes infinitely into the past. Not only does this start to go outside science and into metaphysics, but even the metaphysics has a problem.

And that's because an actually infinite number of past events to create those universes is impossible. You will run into paradoxes. For example, if there were an infinite number of past events, how did we ever catch up to the present? How did the multiverse actually "count" to infinity? On the other hand, if an infinite time has past, why is it now the present - why hasn't everything already happened yet? For more on this, you can look into Hilbert's Hotel Paradox for the problems with actual infinity.

As the eminent mathematician David Hilbert said: "The infinite is nowhere to be found in reality.  It neither exists in nature nor provides a legitimate basis for rational thought. The role that remains for the infinite to play is solely that of an idea."