The way the complicated orbital mechanics work, there is a "gate" which is the only place where asteroids/comets/whatever can cross Jupiter's orbit. This doesn't usually result in them hitting Jupiter, but it does limit their options for hitting Earth.
Been a while since I watched the video, so I don't remember all the details, but that should be the basic gist.
It's not really about "attracting" asteroids, it's more about their orbital trajectory happening to intersect the Earth's at the right time. I think the Moon's gravity is about as likely to redirect an asteroid towards the Earth as it is away, but the Moon also physically intercepts some asteroids, so the net effect would be a slight decrease in Earthly asteroid impacts.