What you are looking for is called "DAW Mode" and I would recommend to read the StudioLive_Series_III_Studio_One_DAW_Control_Addendum_EN_V3_25032019, which you can find in the download area of your mixer.
What you need to do technically is to connect mixer and computer/Studio One in a dedicated network, because DAW Mode does not work via USB, but via UCNET/MIDI. Once you see your Studi One in the UCNET menue of your mixer (scroll down in the list, it's the last and hidden), you can press the DAW button and the mixer will be the controller for Studio One.
One more note: DAW Mode is one of the two ways how to use a StudioLive mixer and Studio One. The second way is the "normal" way: Connect mixer and computer via USB, create a new song in Studio One and load the template for your mixer, then confirm the settings. Now all tracks are routed 1:1 to the channnels of your mixer, so you can start recording. For playing back recorded tracks to the mixer you need to change the input source of the channels from Analog to USB.