Feedburner, http://www.feedburner.com/, offers a service to help you generate nice looking little buttons (a.k.a. chicklets) that aid in your users subscribing to your feed. The easier it is for people to subscribe, the more likely it will happen and the more subscribers you will have. Of course not everybody is going to subscribe it simply increases the odds they will do so.
It's pretty simple to use. In your account there is a page under Publicize named Chiclet Chooser which generates HTML code for each chiclet. To generate a chiclet for a given service, click on the radio button next to that service and in the box below will appear the HTML for that chiclet. Next copy that HTML onto your website and voila you're done.
If you look carefully the feed URL which Feedburner has the subscription for is the feedburner version of the URL. For example on my http://www.7gen.com/technosanity/ podcast they generate this URL: http://www.newsalloy.com/?rss=http://feeds.feedburner.com/TechnosanityPodcast ... that is it's telling the newsalloy.com service to subscribe the feedburner version of my URL.
What if I want to abandon feedburner? Read It's your RSS feed, not Feedburners, and how to maintain the relationship with your readers through your RSS feed for a discussion of the issue. These services like Feedburner want you to have your subscribers subscribed through their service, that way it encourages you to stay with their service. Hence the Feedburner chicklet creator gives the Feedburner subscription URL rather than your own.
Assuming you've done the right thing and set up a soft redirect from your feed URL to Feedburners feed URL, here is the steps to successfully change the chicklet code.
Clicking on the radio buttons gives you HTML code for the corresponding chicklet button. At the bottom of the page is a box containing this code, and you are expected to copy that code and paste it into your website template. Go ahead and do so but there is a second step to perform. Within the template will be one or more instances of a URL named http://feeds.feedburner.com/MYFEEDBURNERID. For each such instance replace the feedburner URL with your own, but do so very carefully.
That's it. It is really simple. Good luck.