What does "Enable Quick Website Search" do on OS X?

If you develop a website you can add a file called an OpenSearch specification that identifies the site as having searchable content.

You can see the apple.stackexchange.com OpenSearch specification here. The important bit is the following line that tells browsers the format for submitting search queries to the site.

<Url type="text/html" method="get" template="https://apple.stackexchange.com/search?q={searchTerms}"/>

The first time you visit a site with an OpenSearch specification Safari adds it to that list you can find under "Manage Websites". So if you start typing into the search bar something like "apple chrome hanging", Safari will see that you have apple.stackexchange.com in your list of quick searches and offer Search apple.stackexchange.com for "chrome hanging".

Nothing to do with Siri or Apple's servers. It all happens on your own machine. It's just a way to submit searches directly to other websites instead of having to open the particular site, click in their search field and type your query there.

You can find Safari's store of Quick Website Search information in /Users/YOURUSERNAME/Library/Safari/SearchDescriptions.plist.