UPDATE: Google has disabled inline installation feature for Chrome extensions.
The documentation: https://developer.chrome.com/webstore/inline_installation
First off, for this to work, you will need to verify the website as something you own via Webmaster Tools.
Verified site requirement
For security reasons, inline installations can only be initiated by a page on a site that is verified (via Webmaster Tools) as being associated with that item in the Chrome Web Store. Note that if you verify ownership for a domain (for example, http://example.com) you can initiate inline installation from any subdomain or page (for example, http://app.example.com or http://example.com/page.html).
Once you verified your site, you need to edit your extension listing via Developer Console to select that the extension is associated with your (verified) site:
If you have done all that, added the link tag and triggered the install()
function (which, if I remember correctly, has to be triggered by a user action, like a click handler), the user will be presented with a small dialog describing your extension, something like that:
There, they can accept or deny without opening Web Store. You cannot silently install an extension, but you can save the user the trouble of going to the Store listing.