All you need to do is use the mouse to select a word (like a band name or album title) from the text you're reading in the browser, then push that Spotify button that appears on the navigation bar after installation.
In just a few seconds a new window will show you all the search results. Click a result to open it up in Spotify and get to listening.
Spotify Chrome Extension is relatively useful, keeping in mind that the time it actually saves you in comparison with manually opening Spotify, entering the name, and hitting play is negligible. Still, it could be helpful for lazy or forgetful users.