In case you are not interested in running a Process Monitor Tool and copying the media file from the temporary internet cache, I wrote a quick and dirty bookmarklet to download the media from within the browser itself. Installation of the bookmarklet is quite easy, make sure that the Bookmarks Toolbar is visible, drag the (below) link to the bookmark toolbar or right click on the link and select "Bookmark This Link" and select "Bookmark Toolbar" in the "Create In" dropdown. Once installed, once you are on a particular media file page on esnips; which you would like to download, just click on the bookmark link.
Fine Print: The bookmarklet is only meant for esnips.com and has only been tested on Firefox, I will try to test it out on IE7; once I log on to Vista sometime tomorrow morning.
Tested it out on Vista using both Fx and IE7 and realized that due to the plugin, the media will start playing within the browser itself after using the bookmarklet. You can then either, use "Save Page As" from the file menu or use the below bookmarklet. This bookmarklet will add a "Download Media" link to the page just below the "Tags"; right click on this link and select "Save Target As..." on IE7 or "Save Link As" on Fx.
Doing a "Save Target As.."
The In-Place Bookmarklet:
(esnips)Add Download Link