mysql after insert trigger which updates another table's column

Try this:

DELIMITER $$
CREATE TRIGGER occupy_trig
AFTER INSERT ON `OccupiedRoom` FOR EACH ROW
begin
       DECLARE id_exists Boolean;
       -- Check BookingRequest table
       SELECT 1
       INTO @id_exists
       FROM BookingRequest
       WHERE BookingRequest.idRequest= NEW.idRequest;

       IF @id_exists = 1
       THEN
           UPDATE BookingRequest
           SET status = '1'
           WHERE idRequest = NEW.idRequest;
        END IF;
END;
$$
DELIMITER ;

With your requirements you don't need BEGIN END and IF with unnecessary SELECT in your trigger. So you can simplify it to this

CREATE TRIGGER occupy_trig AFTER INSERT ON occupiedroom 
FOR EACH ROW
  UPDATE BookingRequest
     SET status = 1
   WHERE idRequest = NEW.idRequest;

Maybe remove the semi-colon after set because now the where statement doesn't belong to the update statement. Also the idRequest could be a problem, better write BookingRequest.idRequest