The new side-B is a big part of it. It's fast and it travels a long distance, making it great for pushing enemies off the side. (Charizard already had up-B as an early kill option, but that doesn't link as well.) Since it's repeatable and can snap to the ledge, it's equally good for recovering. Finally, it has armor around halfway through, giving Charizard a way to approach through projectiles.
Speaking of recovering, Charizard has four midair jumps, a glide that accelerates to ridiculous speeds, side-B, and an armored up-B that goes higher than Mr. Game and Watch's.
The new fair may not do much damage, but it has high base knockback, making it another early kill option. You have to go offstage to kill with it, but Charizard can go super deep without any trouble. Plus, it's a large hitbox that lingers and stays strong the whole time.
Then there's Stealth Rock, one of the better stage control tools in the game. Charizard can throw opponents towards the rocks to extend its combos, or it can DI towards the rocks to save itself from a follow-up. If Charizard's opponent reflects the rocks, Charizard can just make a new set, instantly destroying the reflected rocks.
Finally, nair comes out on frame 4, dealing 15% damage and decent knockback, with range comparable to bair. You can use it out of shield, you can use it as a reverse aerial rush, and you can use it to get back onstage. (Drop off the ledge, turn around by jumping away from the stage, then nair to hit anyone waiting above the ledge. This isn't 100% safe, but it's far safer than fair.)