Google is developing 'Stories' to make web search results fancier4
The AMP extension in question is simply called Stories. These are essentially news-related visual experiences meant to be displayed in the Google search results. They're designed to be visually rich, featuring text and pictures, as well as video and animation. User interaction will be conducted through taps, which would take you from one portion of the story to the next - much like how you flip through Instragram Stories and Facebook Moments. Most importantly, Google AMP Stories are intended to be engaging and fast, sparing the reader from lags and loading circles.
AMP Stories are currently at an experimental stage, and only a handful of publishers have access to the feature. However, it will become available to any publisher in the future. Until then, here's a video of what AMP Stories should look like.