First, Meta Knight pulls out his cape. If he catches a character, the character will be paralyzed. The screen will then fade to black, freezing everything. Meta Knight then says "Behold..." and hits the trapped character(s) and any others who are in range with a powerful slash that deals 42% damage, cancelling out the gray effect and launching fighters who were hit.
While the move is created for the Super Smash Bros. series, Galaxia is the name of Meta Knight's sword, which originated in the anime, Kirby: Right Back at Ya!. However, the name wasn't reffered as such in any of the Kirby games before Brawl. Kirby's Epic Yarn, released after Brawl, became the first to use the name "Galaxia" for the sword. The actual attack itself later appeared in Kirby: Planet Robobot, as part of Meta Knight's moveset, although with a horizontal slash instead of diagonal.
If the user attacks quick enough, they can cancel all the knockback of the move with another attack.
On the ground at high percentages the move will glitch and bring you to your standing position, while you fly in the air. This only happens at higher percentages, and is therefore a KO unless a wall blocks you, or you momentum cancel extremely quickly.
Even if a wall blocks you, you'll be stuck on it at "high speed" for at least a full second, giving the Meta Knight user enough time to charge up a smash or rack up damage.