This is more of an issue with the way that Facebook share works than Contao.
When you share to Facebook you share the link to the page. Facebook will just grab the first bit of text it finds on this page, which in your case is the header text. To change this behaviour you need to tell Facebook which text you'd like it to use when sharing the page.
You can do this by using a meta tag like this in your page:
Code:
<meta property="og:description" content="Article Description." />
When you share the page Facebook will now use this instead of the first text that it finds.
See the URL below for a more detailed look at this. You will probably mostly be interested in the section titled 'Creating a social snippet'.
http://webdesign.tutsplus.com/articl...cial-snippets/
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