There are two separate problems here: the first is parsing out the time from your .start_time input, and the second is incrementing it to be an hour later.
The first is really a string-manipulation exercise. Once you have parsed out the pieces of the string, e.g. via a regex, you could either turn them into a Date instance and use setHours, or you could just manipulate the components as numbers and them reassemble them into a string in the format you desire.
An example of this might be as follows:
var TIME_PARSING_REGEX = /([0-9]{2}):([0-9]{2}):([0-9]{2})/;
function padToTwoDigits(number) {
    return (number < 10 ? "0" : "") + number;
}
$(".start_time").change(function () {
    var stringTime = $(this).val();
    var regexResults = TIME_PARSING_REGEX.exec(stringTime);
    var hours = parseInt(regexResults[1], 10);
    var newHours = (hours + 1) % 24;
    var newHoursString = padToTwoDigits(newHours);
    var minutesString = regexResults[2];
    var secondsString = regexResults[3];
    var newTimeString = newHoursString + ":" + minutesString + ":" + secondsString;
    $(".end_time").val(newTimeString);
});