Not much happens, as the metal shell of the plane acts as a hollow conductor, and the lightning just continues on its way to the ground.
But why doesn't the gigantic amount of current, which is in the neighborhood of 20,000 amps for a typical lightning bolt, harm the passengers inside the aircraft? Because the hull of the plane forms a Faraday cage! A Faraday cage is a hollow shell made of conducting material. A strong electric field outside the cage will force the charge in the material of the cage to redistribute itself, but the interior space inside the cage remains uncharged.
It's still neat to watch though. But that's not why I posted this. I posted this because how did the person filming know the plane was going to get hit by lightning? Here, I'll give you a hint -- sorcery.