What is an .axd file?

Those are not files (they don't exist on disk) - they are just names under which some HTTP handlers are registered. Take a look at the web.config in .NET Framework's directory (e.g. C:\Windows\Microsoft.NET\Framework\v4.0.30319\Config\web.config):

<configuration>
  <system.web>
    <httpHandlers>
      <add path="eurl.axd" verb="*" type="System.Web.HttpNotFoundHandler" validate="True" />
      <add path="trace.axd" verb="*" type="System.Web.Handlers.TraceHandler" validate="True" />
      <add path="WebResource.axd" verb="GET" type="System.Web.Handlers.AssemblyResourceLoader" validate="True" />
      <add verb="*" path="*_AppService.axd" type="System.Web.Script.Services.ScriptHandlerFactory, System.Web.Extensions, Version=4.0.0.0, Culture=neutral, PublicKeyToken=31bf3856ad364e35" validate="False" />
      <add verb="GET,HEAD" path="ScriptResource.axd" type="System.Web.Handlers.ScriptResourceHandler, System.Web.Extensions, Version=4.0.0.0, Culture=neutral, PublicKeyToken=31bf3856ad364e35" validate="False"/>
      <add path="*.axd" verb="*" type="System.Web.HttpNotFoundHandler" validate="True" />
    </httpHandlers>
  </system.web>
<configuration>

You can register your own handlers with a whatever.axd name in your application's web.config. While you can bind your handlers to whatever names you like, .axd has the upside of working on IIS6 out of the box by default (IIS6 passes requests for *.axd to the ASP.NET runtime by default). Using an arbitrary path for the handler, like Document.pdf (or really anything except ASP.NET-specific extensions), requires more configuration work. In IIS7 in integrated pipeline mode this is no longer a problem, as all requests are processed by the ASP.NET stack.

from Google

An .axd file is a HTTP Handler file. There are two types of .axd files.

1. ScriptResource.axd
2. WebResource.axd

These are files which are generated at runtime whenever you use ScriptManager in your Web app. This is being generated only once when you deploy it on the server.

Simply put the ScriptResource.AXD contains all of the clientside javascript routines for Ajax. Just because you include a scriptmanager that loads a script file it will never appear as a ScriptResource.AXD - instead it will be merely passed as the .js file you send if you reference a external script file. If you embed it in code then it may merely appear as part of the html as a tag and code but depending if you code according to how the ToolKit handles it - may or may not appear as as a ScriptResource.axd. ScriptResource.axd is only introduced with AJAX and you will never see it elsewhere

And ofcourse it is necessary