You didn't used to have to have a browser extension, not for sites that participated. So for instance, I had flatr configured to follow all of my YouTube Likes...
The extension uses a lot of data points (including your browser history) to decide what to flattr. But we never get the full data, the algorithm runs in the browser and only the results (the flattrs) are sent to us. This is the most privacy friendly solution possible that still gives enough data for us to know who should get the money and for you to know what you flattred. As well as for creators and publishers to know what content actually generated the money.