When an engine fails you are using the rudder to create an aerodynamic force to counteract the yaw from asymmetric trust. To create more force you either need larger deflection or higher airflow/speed.
Propellor slipstream can create more drag when flaps are deployed, the increased drag would be on the live engine side with one engine out. That drag would counteract the yaw from asymmetric trust, therefore you need less rudder or speed to create the same force. So with one engine and take-off flaps the asymmetric force is less, than with flaps up.
As rudder deflection is limited by it's stops, when an engine fails with flaps and you use full rudder you would create the required force at a lower speed. When you raise the flaps, you create more asymmetry, with the rudder at the stop only way of creating more force is by increasing speed.