It rewards those who put the time in to learn the inner-workings of the game. And this isn't just for longer &/or more damaging combos or links, but also the neutral game (I.E. what you're doing when you're not attacking or blocking). It doesn't allow just anyone to pick up the game & start winning, and encourages people to actually play the game & learn along the way. That being said, as far as casuals are concerned, the game also has to still be fun without reaching that competitive level for people to get into it in the first place. And in certain cases, the competitive scene may increase the execution barrier by discovering exploits that change the metagame. Melee is a prime example of this, namely with the discovery of techniques like wave-dashing & L-Cancelling. Despite Melee's high execution barrier for competitive play, it's still an insanely fun Smash game for casual play. Even Smash 4 is starting to develop techniques that could change the metagame, like the recently-discovered Bidou exploit. However, we don't know if Bidou catch on yet, specifically due to its unorthodox control set-up requirements & the fact that it can't be used as efficiently with the GameCube controller (the controller of choice for competitive Smash). But in the case of Street Fighter III: Third Strike, while the execution barrier is there, it may not be as fun for a casual player to just pick it up & play (then again, I've been playing fighting games since I was 3, so I may have a bias in this regard).
In short, execution barriers give a larger pay-off to those who play the game a lot. But at the same time, games with high execution barriers for competitive play can also be fun without reaching that level of competitive-ness.