Anti-Support fleets generally protect the main fleet of damage dealers from outside interference from ships such as Stealth Bombers, Electronic Attack Frigates, Interceptors, Interdictors, and Tactical Destroyers.
Remember when you were a newbie tackler on your very first op? There you were, burning towards the enemy fleet, and then BANG you were dead. Anti-Support did that. Anti-Support is largely Tier 3 Tactical Destroyers, Tier 2 Battlecruisers and Heavy Assault Cruisers who are set up to kill anything that isn't a battleship.
Anti-Support kills Interdictors, stealth bombers, interceptors, ECM ships, basically ships that, while small and fragile, can really screw over the fleet. Traditionally, Anti-Support is specifically set up for fast target locks, good DPS, and excellent tracking of small and medium targets at ranges from 50-100KM.