This scenario illustrates a classic trigger recursion issue triggered by the Salesforce Order of Execution. When a record is updated, Salesforce runs through a specific sequence: first, it executes "Before-Save" Record-Triggered Flows, then "Before" triggers, followed by "After" triggers, and finally "After-Save" flows and processes. If a process or flow updates the same record that initiated the transaction, the entire cycle of Apex triggers can be re-invoked within that single transaction.

To prevent logic from running multiple times during these re-entrant cycles, developers implement a static boolean variable as a "recursion guard." Static variables in Apex persist for the entire duration of a single transaction. By checking the value of a static boolean at the start of the trigger, the code can determine if it has already processed the current set of records.

Option B is the correct implementation of this pattern. It defines `public static Boolean firstRun = true;` in the handler class. The trigger checks if `firstRun` is true, executes the handler logic, and then immediately sets `firstRun` to false. Any subsequent execution of the Case trigger within the same transaction will find the variable set to false and skip the increment logic.

Option A is incorrect because it resets `firstRun` to true every time the trigger starts, defeating the guard. Option C incorrectly uses an instance variable (non-static), which is recreated for every call. Option D uses a local variable within the trigger body, which is re-initialized to true every time the trigger fires, providing no protection against recursion.