The issue you are having is not due to anything wrong with your code, but with security policies of modern browsers. Here's what happens on your development machine:
Your browser loads your local HTML file.
Your browser executes the javascript, which tries to access a file on your machine.
Your browser says, "NO!" Because this is a huge security error - without this policy websites could read through every file on your hard drive or silently send copies of any of your private information to their servers, just because you visited a site with javascript enabled. BAD!
There are some ways to try to tell your browser "No, it's ok, I want to allow this..."...but you know, this has become exceedingly difficult as it often silently breaks with new browser versions. I've slammed my head against the wall way too often, so I might suggest you skip trying to make your browser OK with what you are trying to do.
Now, why does this work on a live site? Here's what happens.
Your browser loads a website.
Your browser executes the javascript.
The script asks for a file to be loaded/accessed from a website.
Your browser says..."well, we're already on this website, so sure! Load all the files you want from that web server!" And your browser kindly gets the file, and returns it to your script, where you can painlessly include the HTML to your hearts content.
To make this work on your development machine, you have ultimately 3 choices:
1) Upload the files to a web server, then do your testing there.
2) Make your own "localhost" web server. You then access your site with something like localhost/index.html - etc. This is just enough to prevent the browser from shutting down your file load requests, because you are requesting an HTTP operation, note a FILE operation.
3) Try to force your browser to allow these requests. Details vary by browser, some browsers won't let you do this at all, and I've given up on doing this myself.
The hidden 4th choice is using HTML5 File System features, but with such poor support for technology I suggest you not even try it - the bug you are facing is purely with your development machine, so changing the technology you are using purely for a minor development convenience seems silly.
Severin provides links to the excellent XAMPP and MAMP software packages, which are the easiest way of getting yourself a good development localhost server.