The line that sends the email is
$mresult = mail($toAddress, $subject, $emailMessage, $fromAddress);
and [man]mail[/man] tells us that the third parameter corresponds to message to be sent. In this case, you provide the variable $emailMessage for this purpose, so let's look at what you assign to this variable in the code above.
$emailMessage .=
$name = $_POST["name"];
That statement assign the value of $_POST['name'] to $name, and then concatenate the same value to $emailMessage. Do note that unless you've left out code, $emailMessage has not yet been defined, which means that concatenating things to it will result in an error (check your error log), since the operator .= is the same as writing
$emailMessage = $emailMessage . 'stuff to append';
and if $emailMessage doesn't yet exist, you can't access it's value. Hence the error "undefined variable: emailMessage".
After that, you do not do anything else with $emailMessage. You do however assign the value of $_POST['message'] to $message, but that isn't the same variable as $emailMessage...