This short tutorial from Ableton covers Start/Stop Sync, an optional feature within Link that goes beyond tempo alignment. Once enabled, pressing play or stop on any device in the Link session triggers the transport on all connected devices simultaneously.
To activate it, open Live's preferences, go to the Link MIDI tab, and turn on Start/Stop Sync. The same setting also needs to be enabled on each connected device, so it's a two-sided switch worth keeping in mind when troubleshooting.