First, define a JavaScript function that will be called after the event is fired. Don't forget that this event will receive three values as parameters: start (Date object), end (Date object), and column (string).

I expect that you need to prefill <input type="text"> elements. If these elements are generated using ASP.NET controls (e.g. TextBox) you need to ensure that you use the right IDs (you get it by using TextBox.ClientID property).

<script type=" type='text/javascript">
function Prefill(start, end, column) {
 document.getElementById('startField').value = start.toLocaleString();
 document.getElementById('endField').value = end.toLocaleString(); 
}
</script>

Remember that the start and end parameters are Date JavaScript objects so you can use its methods to extract the necessary information.
This will work if you have HTML similar to the following in your page:

<input type="text" id="startField">
<input type="text" id="endField"> 

Then set DayPilotCalendar.TimeRangeSelectedJSFunction to "Prefill" and it should work.

I'm working on a documentation where all this client-side stuff will be explained. This example is probably interesting for more people so I will it in the demo - http://www.daypilot.org/demo/.

Let me know if this doesn't work for you.

Please note that I have simplified the API in DayPilot Pro 3.5 a little bit so now instead of

DayPilotCalendar1.TimeRangeSelectedJSFunction = "function(start, end, column) {document.getElementById('" + DetailsView_AddAppointment.FindControl("TextBox_StartTime").UniqueID + "').value = start.toLocaleString(); document.getElementById('" + DetailsView_AddAppointment.FindControl("TextBox_EndTime").UniqueID + "').value = end.toLocaleString();}";

you should use:

DayPilotCalendar1.TimeRangeSelectedJavaScript = "document.getElementById('" + DetailsView_AddAppointment.FindControl("TextBox_StartTime").UniqueID + "').value = start.toLocaleString(); document.getElementById('" + DetailsView_AddAppointment.FindControl("TextBox_EndTime").UniqueID + "').value = end.toLocaleString();";

There are two changes:

1. TimeRangeSelectedJSFunction was renamed to TimeRangeSelectedJavaScript.
2. The parameter is now not a JavaScript function but directly a JavaScript code. The same variables are available: start, end, column.