I imagine they are either being stored in another (non accessible index) or being compiled when the query runs. (My guess would be for high volume non time sensitive queries they are cached elsewhere, and for low volume and newsy type queries they are generated in real time.) In any event as far as I know there's no publicly accessible link you could view these from.
I do know what they look at to generate them so here's that:
Google chooses between the pages meta description tag the body text on the page.
The description tag is often not implemented properly - for example the same will be used across an entire site. It looks like this
<meta name="description" content="page summary content goes here">
Google will look at how relevant the description tag is and how relevant the body text is to the search query and choose between them, often combining two snippets of page content or combining some of the meta description with some of the body text (one such example is shown below):
Typically Google will use the meta description tag in it's entirety for homepages as that is a websites branded summary of their company, but for deeper pages they prioritize body content.
Google states that they do reference the open directory project, but I would be surprised if this info was still weighted very highly.
Google will also create rich snippets based on specified markup in a page, you can see that in action with the star ratings in the image above. More info on rich snippets is here: https://support.google.com/webmasters/answer/99170
The meta description tag should be around 155 characters. http://moz.com/learn/seo/meta-description
And for completeness there are also sitelinks which are generally only given if the root domain has enough domain authority: https://support.google.com/webmasters/answer/47334?hl=en