getting the value of daterangepicker bootstrap

$('#IDOfDateRangePicker').data('daterangepicker').startDate;
$('#IDOfDateRangePicker').data('daterangepicker').endDate;

Give your button an id

<button type="button" id="saveBtn" class="btn btn-primary" data-dismiss="modal">Save changes</button>

Add startDate and endDate variables to your script

var startDate;
var endDate;

Set these variables in the daterangepicker callback

startDate = start;
endDate = end;

Wire up the click event for that button in your $(document).ready function

$('#saveBtn').click(function(){
    console.log(startDate.format('D MMMM YYYY') + ' - ' + endDate.format('D MMMM YYYY'));
});

The complete javascript

<script type="text/javascript">
var startDate;
var endDate;

$(document).ready(function() {
    $('#reportrange').daterangepicker(
       {
          startDate: moment().subtract('days', 29),
          endDate: moment(),
          minDate: '01/01/2012',
          maxDate: '12/31/2014',
          dateLimit: { days: 60 },
          showDropdowns: true,
          showWeekNumbers: true,
          timePicker: false,
          timePickerIncrement: 1,
          timePicker12Hour: true,
          ranges: {
             'Today': [moment(), moment()],
             'Yesterday': [moment().subtract('days', 1), moment().subtract('days', 1)],
             'Last 7 Days': [moment().subtract('days', 6), moment()],
             'Last 30 Days': [moment().subtract('days', 29), moment()],
             'This Month': [moment().startOf('month'), moment().endOf('month')],
             'Last Month': [moment().subtract('month', 1).startOf('month'), moment().subtract('month', 1).endOf('month')]
          },
          opens: 'left',
          buttonClasses: ['btn btn-default'],
          applyClass: 'btn-small btn-primary',
          cancelClass: 'btn-small',
          format: 'DD/MM/YYYY',
          separator: ' to ',
          locale: {
              applyLabel: 'Submit',
              fromLabel: 'From',
              toLabel: 'To',
              customRangeLabel: 'Custom Range',
              daysOfWeek: ['Su', 'Mo', 'Tu', 'We', 'Th', 'Fr','Sa'],
              monthNames: ['January', 'February', 'March', 'April', 'May', 'June', 'July', 'August', 'September', 'October', 'November', 'December'],
              firstDay: 1
          }
       },
       function(start, end) {
        console.log("Callback has been called!");
        $('#reportrange span').html(start.format('D MMMM YYYY') + ' - ' + end.format('D MMMM YYYY'));
        startDate = start;
         endDate = end;    

       }
    );
    //Set the initial state of the picker label
    $('#reportrange span').html(moment().subtract('days', 29).format('D MMMM YYYY') + ' - ' + moment().format('D MMMM YYYY'));

    $('#saveBtn').click(function(){
        console.log(startDate.format('D MMMM YYYY') + ' - ' + endDate.format('D MMMM YYYY'));
    });

 });
</script>  

var endDate=  $("#IDOfDateRangePicker").data('daterangepicker').endDate.format('YYYY-MM-DD');

or other formats;

.endDate.format('YYYY-MM-DD'); 

.endDate.format('D MM YYYY');

.endDate.format('D MMMM YYYY');