To supplement KRyan's answer with the detailed mechanics...
In return, they provoke attacks of opportunity from you each time they swing.
Since you mention "sufficient Dex", you are probably aware of it, still I find it worth mentioning: Robilar's Gambit only gives the character an opportunity to strike back. This means that:
- the character may choose NOT to strike back
- the character may be unable to strike back: the opponent may be out of range or your character may be out of Attacks of Opportunity for example
Normally, a character has only one Attack of Opportunity per turn; specific feats such as Combat Reflexes increase this amount by your Dex modifier.
Resolve your attack of opportunity after your foe’s attack.
The Attack of Opportunity of your opponent is taken during your movement, while you are within range. The Attack of Opportunity you may take must be taken at this moment, in this spot.
Also, obviously, you may only counter-attack if you are still alive and conscious...
In parting, I also feel the need to specify that the foes are not forced to make their own Attacks of Opportunity against the passing character. Whilst less intelligent foes (animals) might snap at the first opportunity, a more intelligent one may prefer to reserve its Attack of Opportunity for someone else or realize that the retaliation that comes after such an attack is not worth it.
Furthermore, certain feats allow taking other actions on an Attack of Opportunity. For example the feat Evasive Reflexes (Tome of Battle, p. 30) allows the character to take a 5 ft. step instead of making an Attack of Opportunity. This will not trigger Robilar's Gambit.