good call; this is a question that should probably have some documented answer to it. the guide is horribly outdated, so i'll put this here for the time being.
you can't airdash from a standstill. airdash incorporates momentum in some way. there are a couple ways to do this:
1. be running or otherwise moving, and jump while strafing to the left or right.
2. the more spoken-of method when people refer to airdashing - jumping immediately upon landing from a previous jump and changing the direction of your momentum with the control stick (and L, if you're on Single). this allows a flurry of multidirectional movement without having to be on the ground for extended periods of time.
the reason for this is PROBABLY (SFA's movement system is as of yet unplumbed) that SFA uses a jump-queuing method like Quake or similar FPS games: when you push the jump button, it sends "jump" to the GCN once, and only once, and executes that jump as soon as the player lands (within limits; you can't immediately jump after falling from a great height).
therefore, if you tap jump exactly when or a very little before you land, you will immediately jump, within a minimum amount of frames of being on the ground. this means that in that very small time frame of being on the ground (but not actually standing or walking) you can input direction from the control stick, and have your character respond accordingly.
in other words, by chaining together a whole ton of these jumps, you can switch direction basically at will. this has huge applications in SFA movement, because moving unpredictably is otherwise pretty hard.
i'm trying like hell to look for an in-game example of it but i honestly can't find anything, which is kind of dumb! uploading a tutorial or demonstration of this has been on my list of stuff to do for a long time, so these instructions'll have to do for now:
- walk in some direction, holding L (if using Single)
- jump, holding the control stick in the same direction in which you were walking
- immediately upon landing, push the control stick in a different direction and tap Y
- repeat from step 2 as necessary