it's not allowed to concatenate strings in a function (the Dot operator).
Try this:
<?php
$the_date = date("d-m-Y", mktime (0,0,0,date("m"), date("d"), date("Y")));
$new_date = date("d-m-Y", mktime (0,0,0,date("m"), date("d")+$days_to, date("Y")));
echo "The Date: $the_date<br>";
echo "<form type='POST' method='POST' action=''>Days to: <input type='text' name='days_to' value='$days_to'>\n";
echo "<input type='submit' name='Submit' value='check'></form>";
echo "New Date: $new_date";
?>