仅将选定的复选框放入PhP变量中，然后在下面显示该变量

I want to get selected values (of which multiple selections are possible) from my database generated dropdown menu and store those into a PhP variable. I then wish to display the content of that variable into a simple div element below.
This is my code so far which results in nothing inside my simple div:

<form id="menu1" method="POST">
        <h2>Area Code</h2>
        <select id="multi-select1" name="multi_select1" multiple="multiple">
            <?php
                //The query asking from our database
                $areaCodeSQL = "SELECT ac.Number AS `AreaCode`, ac.Name AS `AreaName`
                                FROM `AreaCodes` ac";                                                               //SQL query: From the table 'AreaCodes' select 'Number' and put into 'AreaCode', select Name and put into 'AreaName'

                $areaCodeResults = $conn->query($areaCodeSQL);                                                      // put results of SQL query into this variable

                if ($areaCodeResults->num_rows > 0) {                                                               // if num_rows(from $results) is greater than 0, then do this:
                    // output data of each row
                                foreach($areaCodeResults as $areaCodeResult)                                        //for each item in $areCodeResults do this:
                                    {
                                        $areaNameAndCode =  $areaCodeResult['AreaCode'] ." ". $areaCodeResult['AreaName'];  //get AreaCode and AreaName from query result and concat them
                                        $areaName = $areaCodeResult['AreaName'];                                    // get AreaName
                                        $areaCode = $areaCodeResult['AreaCode'];                                    //get AreaCode

                                        ?><option class="menuoption1" name="menuAreaCode" value="<?php echo $areaCode ?>"  ><?php echo $areaNameAndCode; ?></option><?php  //Create this option element populated with query result variables
                                    }
                } 
                $result = $_POST['multi_select1'];                   
            ?>
        </select>   
    </form>

 <div id="showResults1"><?php echo $result ?></div>

Looking around online suggests I might need to use AJAX and jQuery but my tutor buddy insists this can be done within this one script. But I have no idea why my attempt does not work, can some one point me in the right direction???? :-)