I just realized what's going on with the two selects on that page haha. Let's say we were going to insert these values into the database. You would store all the values in variables and then insert it into the query which inserts a row in the database. The fields in your first code don't match up here so I'm making this up.
This goes in submit_add_tenant.php:
$forename = mysql_real_escape_string($_POST['tenForename']);
$surname = mysql_real_escape_string($_POST['tenSurname']);
$tenSuitablePostcodes = mysql_real_escape_string(implode(',', $_POST['tenSuitablePostcodes']));
$sql = "INSERT INTO table (`tenForename`, `tenSurname`, `tenSuitablePostcodes`) VALUES ('$forename', '$surname', '$tenSuitablePostcodes')";
$res = mysql_query($sql);
Your second select would need to be renamed into
<select id="box2View" multiple="multiple" size="3" name="tenSuitablePostcodes[]">