Well, I think the reason this is of interest is because at the moment, players of the 3DS version have no C-Stick. This quarter-circle input lets you fulfill the role of the C-Stick in Brawl for instant dash attacks without, well, a C-Stick.
Also, a down input on the C-Stick wasn't really coded to specifically execute a dash attack when running. In Brawl, the C-Stick was coded so that an input registered as (tapped direction + standard attack) at once, if the stick was set to smash. So pressing forward on the C-Stick gave (forward tap + attack), which naturally gives an Fsmash. Since you can jump out of a run/dash, pressing up on the C-Stick counts as a jump + attack. The jump is "cancelled" immediately by the attack input, which results in an Usmash right out of your dash.
However, Brawl doesn't let you cancel your run with a crouch. So when you press down on the C-Stick while running, the (down tap) component of the input is ignored by the dash, and only the attack input goes though. And as we all know, pressing the attack button during a dash/run results in a dash attack. This is also why certain DACUS were easier to perform with just the C-stick. Flicking it down the immediately up while running gives you a dash attack first, which is then cancelled by an Usmash. Iirc this is one of the easier ways to DACUS with Snake
I got carried away