I wanted to learn which parts of the easy discuss is picked up by the google crawler.

By default, all the Easydiscuss page will allow Google to index it, unless if you do not want certain page to indexed by Google, then you have create menu item for it.

Is there a heading and meta description that the google search engine gets to see? Which are these?

If I am not wrong Google will look for the <title> tag and <meta name="description"> tag so it will populate the search result title and description.

For example :
Single discussion page
<title> - it will show the discussion post title
<meta name="description"> - it will show the first sentence of the discussion content.