Does annihilator trigger before or after the defending player can cast spells?

Annihilator triggers at the beginning of the declare attackers step, the moment the creature with annihilator is declared as an attacker (CR 702.86b). This trigger goes on the stack and must fully resolve before the game moves forward.

Once the trigger resolves, the defending player sacrifices the required number of permanents immediately. Only after that process is complete does the game proceed to the declare blockers step (CR 509.1). So the sacrifice happens well before blockers are ever chosen.

Regarding casting spells: players receive priority after the annihilator trigger is placed on the stack (CR 117.1), so technically both players can respond to the trigger before it resolves. However, the defending player cannot prevent the sacrifice by casting spells — once the trigger resolves, the sacrifice is a cost-like effect that happens immediately with no further chance to respond before permanents are gone.

Concrete example: You attack with Emrakul, the Aeons Torn (Annihilator 6). As soon as Emrakul is declared as an attacker, the annihilator trigger goes on the stack. After it resolves, your opponent sacrifices 6 permanents. Only then does the game move to the declare blockers step — your opponent cannot cast spells or activate abilities to save those permanents after the trigger resolves.

This makes annihilator extremely powerful: the defending player loses significant resources before they even get to block, and they cannot use most in-response tricks to avoid the sacrifice once the trigger resolves (CR 702.86, 509.1).