the problem is case sensitivity.
the form element name is "Appointment_request_group_Best_Approximate_Time[]"
<select class="formListfield_Medium" name="Appointment_request_group_Best_Approximate_Time[]" id="Appointment_request_group_Best_Approximate_Time" tabindex="5" multiple="multiple" onblur="hideServerError('Appointment_request_group_Best_Approximate_Time_ServerError');">
but in the echo, you are capitalizing the "r" in "request, so it becomes "Appointment_Request_group_Best_Approximate_Time". the code you have is:
<?php echo((isset($_POST["Appointment_Request_group_Best_Approximate_Time"]))?implode(", ",$_POST["Appointment_Request_group_Best_Approximate_Time"]):"") ?>
but it needs to use a lower case "r" for request:
<?php echo((isset($_POST["Appointment_request_group_Best_Approximate_Time"]))?implode(", ",$_POST["Appointment_request_group_Best_Approximate_Time"]):"") ?>
PHP is case sensitive.