Imagine the device has a (non-encrypted) database, and the app runs locally and interacts with that database, like normal. Think localStorage if you are web-oriented.

The sync would be a separate background process (i.e. managed by the "Solid server" part) that handles encryption and decryption. As for how to manage a "circle" of devices that share the key without revealing it to the server: you can add a device via a key-exchange with the untrusted device asking a trusted device for the key. You can perform a key roll to remove a device. This can all be done automatically, though, where all the user sees is a control to add or remove a device. The hard part is key escrow (you throw all of your devices in a lake), by password protecting a copy of the key. Apple uses HSMs and Signal uses SGX to prevent brute-forcing this backup key.