In it's purest form, Attack speed is simply how often you can swing your primary weapon, or weapon(s) if you're dual wielding. Each weapon has a speed value, and any increase to your attack speed % is a direct deduction from that value. So if you have a weapon that swings every 2 seconds, and you have a 25% bonus to attack speed, it will instead swing every 1.5 seconds.
This means that an attack speed stat has a much greater effect on a weapon that would be a bit slower otherwise. As you are getting a greater "discount" from the normal swing speed, but getting the higher damage typically associated with slower weapons.
But to say that is the only effect of attack speed would not be the whole story. Classes have a resource that is generated by their "primary" attack, whether it be Fury, Spirit, Hatred, etc. So, hitting more often makes more of your resource available, which in turn increases DPS by making those skills available. Character sheet DPS can be very misleading. You should also consider that hitting faster affects how much healing you do from your +life on hit statistic, and any other crowd control effects triggered by your weapon.