Allowing LAN traffic lets you connect to other devices on the same network. This is handy for printing, network file access and the main reason we added it in the first place, Chromecast.
Without this option enabled, your device is protected from other devices on the same network when the VPN is enabled, handy for places like public WiFi in cafes or airports, or if you live with other people you don't know or fully trust but use the same router as them.
Shown on Desktop app - same instructions apply to the Android app*. This option is not available on the iOS app
And that's it! You have enabled Allow Lan traffic. You should now be able to communicate with all other devices on your network.
*Android Settings have an option to "Block connections without VPN". This Setting MUST be disabled if you want to access other devices on your network while still connected to the VPN.