The subscription starts officially when the order is completed, as well as for any other product in the store. Payment modes are the same you set in WooCommerce > Settings.
Among all payment methods available, PayPal is the only one that has been configured in order to detract the subscription recurrent fee automatically until subscription expiry date. In case funds in the concerned PayPal account are not enough, the subscription will be automatically cancelled from the system.
If you set any other payment method, it will be up to customers to remember recurrent payment deadline and send the amount to the admin to keep benefiting from the subscription.