It is a annoying that RSS feeds do not display images correctly, if your feedreader is offline. The problem is, that the links generated by the SharePoint RSS feed api returns relative links instead of absolute links.

But there is a solution: --> Images Broken When Viewing Windows SharePoint Services RSS Feeds in Outlook

Here is a small update if you want to make a regular link absolute as well:

_imagesRegex = new Regex(@"(<(img|a) .*? \s (src|href) \s* = \s* [""']?)/", RegexOptions.IgnoreCase | RegexOptions.IgnorePatternWhitespace);