The short answer is very little.
We effectively query GitHub via their API on-demand. This means that little to no data is stored on our servers. We do ephemerally cache this data to memory (redis) for two minutes for performance reasons. This means if actions you take effectively make the same request to GitHub that we won't keep going back to GitHub for the same data for a short period. This makes the app run faster as there's latency in any over-the-network API call.
The only data that is stored to disc are webhook payloads. If you add an integration into a chat room, every time there is activity on that integration, metadata is sent to our servers and displayed in the activity feed on the right side of the application UI.
We do not read, nor write your source code.