There are two ways to get stories from the site onto an Amazon Kindle:
#1
When you browse the site on the kindle directly, you should get the small screen version of the site which has a 'Download' button. (If you don't see the small screen version of the site and get the full version, then click the 'Settings' link in any listings page, and from the screen size menu, choose 'small' instead of auto or normal.) Clicking the 'Download' button will reveal a kindle button. Click the kindle button to download the story to the kindle.
Method #1 is fairly straight forward and works well.
Method #2 is more involved, but it results in better files on the kindle:
#2:
Kindle files emailed to your kindle directly. To do that, just configure your 'Downloads Preferences with your kindle's email address and configure your Kindle with the site's sender email to allow emails from the site to your kindle.