That is very likely a Flash issue. Currently the site uses Flash to do the large file uploading. It sets a transparent Flash button on top of the "Select Image" button, so that when you click on it, it will open a file dialog box. In the near future, we will switch this to HTML5.
In the meantime, make sure that you install the latest version of Flash from Adobe here:
http://www.adobe.com/go/EN_US-H-GET-FLASH
Check that Flash is actually running. Go to YouTube.com and see if you can watch a video.
Use a modern web browser such as Chrome, Firefox, Safari. If possible, don't use Internet Explorer (not even IE9+).
If that doesn't work, please help me debug further by responding with this:
1. Your Operating system
2. Your Browser
3. Test the site in Chrome. Click on "View" -> "Developer" -> "Javascript Console". Visit the home page. Tell me if there is any message in the console
Thanks,
Leonard