The extension asks this server for the sky above you. To do that the server looks up the approximate, city-level location of your IP address using ip-api.com, then fetches current weather for that location from Open-Meteo. Open-Meteo receives coordinates rounded to 0.1 degrees, never your IP.
The server keeps your IP and its city-level location in a cache for up to one day so it does not have to look it up again, and caches weather per rounded location for ten minutes. Nothing else is stored. No browsing history, no page content, no identifiers.
The extension itself only stores your settings and the latest weather in your browser. It never reads or changes the pages you visit; it draws its own layer on top.
Nothing is sold or shared with anyone beyond the two services named above.