Giving regions in diplomacy is when a city a faction wants can come towards to the faction who sent the deal or is sending the deal, along with this, if you pay a faction, the chances of deals happening will increase. The next log will be about the last two features of diplomacy, Map information and Becoming Protectorate.
The infdev screenshot is from turn 79, where Auricus wants a ceasefire from Starlixia, and then the offer turns to a tribute, and a ceasefire.