First off, I would like to make it clear that Mellee and Brawl are two completely different games. Brawl is not just Mellee with more characters and stages. The developers actualy made it play much slower and removed certain programing which allowed for advanced techniques in Mellee such as wavedashing, dash dancing, L-canceling, and even combos.
Wavedashing, dash dancing, and L-canceling (short for lag canceling) are movement and speed related techniques which cause matches to become extremely fast and heavy in mindgames. You can learn how to perform these techniques yourself here.
The most important thing that makes Mellee competitive is the combos. Yes there are actualy combos in this game. The reason for this is hitstun, the moment after recieving an attack when a character is unable to perform attacks, blocks, or dodges. When an opponent is in hitstun, attacks can be strung together into combos. Unfortunately in Brawl, hitstun was severely lowered which makes performing combos in that game very difficult.