On a classical rental, you have no specific action on Drivy to start a rental. The owner is in charge of providing the contract, you simply need to fill it together, and sign it.
On an Open rental, you'll have access to the exact location of the vehicle 1 hour before the beginning of the rental. Once you're there, go to Your rentals > selected rental then click on "See on Open". Click on "Start the rental". You'll be guided step by step.
In any case, the rental starts automatically at the beginning of the half-day (midnight for a morning rental, midday for an afternoon rental).