It's a combination of a lot of things that make them fast. It takes a lot of practice to play that quickly, and no it is not all wave dashing and wave landing. Actually it's barely any of that a lot of the time. The most important part of playing fast is probably the short hop. That and L-cancelling. Short hop allows you to perform air attacks at a very close point to the ground. L-cancelling is simply hitting the shield button before you hit the ground while you're doing an air attack. It cuts the lag time of landing with the attack in half. Try it with Link's down-air to see its power.
Falco also has a special move - his laser. If you use it on the ground, it is amazingly slow and completely terrible. If you use it in the air, however, the second you land, you are magically out of the entire animation. This allows Falco to fire lasers while moving and at the same time, avoid the lag of firing lasers entirely. Falco also has one of the most powerful moves in the game - his reflector shield (often referred to as "shine"). If you hit them with it (the very beginning of the move hits enemies that are near you), it hits the opponent up, and allows for some of the most devastating combos. However, there are unique properties of this move that make it so powerful.
1) It takes only 1 (I think, I actually can't remember the exact amount, but it's extremely low) frame to hit meaning it's basically instant
2) You can jump out of it
If you can jump, you can wave dash. You can short hop. You can do ridiculous things really fast. These are many of the techniques that falco players use all the time.
Now, one of the biggest things is that once you've mastered all these small, individual techniques, you have to tie them together. It means nothing to L-cancel if you do absolutely nothing with your 5-10 extra frames. You have to get used to the timing of it, and you have to use all of your available frames in order to maximise speed.
What makes them so fast is simply short-hopping lasers or air moves (which are L-cancelled), and doing something the moment you can. The shine is basically a free action if you plan on jumping anyway, so you can throw them out here and there.