以下のようにしてみてください：

AJAX リクエストをchange()使用してそのボックスのメソッドを記述します。select

$('select#AnswerQuestionId').on('change', function() {
   var questionID = $(this).val();
   // send ajax
   $.ajax({
      url: 'admin/answers/add/' + questionID,
      method: 'get',
      dataType: 'json',
      success: function(response) {
          // for example
          // response = [{'Answer': {'id': 1, 'answer': 'ans 1'} }, {'Answer': {'id': 2, 'answer' : 2}}....];
          // now loop over the response
          var html = '';
          $.each(response, function(index, val) {
              html + = '<div id="answer_'+ val.Answer.id +'">'+ val.Answer.answer +'</div>'
          });
          // append this html to target container
          $('#target_container').append(html);
      }
   });
});

CakePHP 側では、次のようにしてみてください。

public function admin_add($id = null) {
    if( !empty($id) )
    {
        $this->Answer->recursive = -1;
        $answers = $this->Answer->find('all', 
                                        array(
                                         'fields' => array('Answer.answer', 'Answer.id'), // assume the answer contains in answer field in db
                                         'conditions' => array('Answer.question_id' => $id)
                                        )
                                      );
        echo json_encode( $answers );
    }
}