For Vmc condition, one of the solutions is to bank up to 5 degrees into the operating engine to increase rudder effectiveness to maintain control. Why is it up to 5 degrees? What happens if the pilot banks more than 5 degrees into the operating engine?
The 5 degrees of bank is to create a side slip component that offsets the skewed thrust line created by the asymmetric thrust, and the rudder input made to counteract the asymmetric thrust.
You have the live engine on one wing that wants to make the airplane turn. You apply opposite rudder to stop the turn. With the rudder moment pushing sideways, you end up with a resultant thrust line that is offset, and the airplane proceeds forward with a lateral skew toward the dead engine even though you think you're going straight.
By banking into the live engine, bank angle makes the airplane want to side slip toward the down wing, which is in the opposite direction to the skew effect mentioned above. The 5 degrees of bank is roughly what gives the necessary amount of side slip tendency. Close enough in other words.
The result is that you will be flying with 5 degrees of bank, but actually proceeding straight through the air. The skid ball will be offset into the bank because you are actually still in coordinated flight and the offset location of the ball is the true "centered" location.
These figures are a regulatory baseline for sizing the ailerons, vertical stabilizer and rudder for an aircraft. The 5° bank limit is done to minimize the load factor on the aircraft while providing a force to counteract rudder input required to maintain a coordinated flight path.
In the event of an engine failure in a non-centerline thrust twin or multi engine aircraft, the operative engine is going to create a strong yawing moment about the vertical axis of the aircraft in the direction of the dead engine. Uncorrected, this results in a forward slip toward the side of the good engine and, when combined with the fuselage blanking airflow over the wing on the dead engine side, a rolling moment also develops about the longitudinal axis in the direction of the dead engine. At low speeds, combined with the high drag created by the slipping condition plus a 50% loss of total available thrust from the engine failure, this can quickly snowball into a departure from controlled flight and crash. The typical action is to apply rudder in the direction of the good engine to counteract this forward slip. However, while the nose will be aligned with the desired flight path doing this, the actual flight path is a side slip towards the dead engine side, which creates excess drag. The only available counter to this is to bank the airplane into the direction of the good engine to counteract the rudder force using the horizontal component of lift. This results in a coordinated flight track parallel to the horizontal axis of the aircraft with a minimum amount of drag.
If excessive bank angle is used to do this, the vertical component of lift is diminished, requiring a greater angle of attack to be imposed upon the wings to stay aloft. This in turn creates more induced drag. The regulations for aircraft design of light twins, therefore, dictated that, in a worst case Vmca, directional control must be maintained with a bank angle NOT GREATER THAN 5°.
Harry Horlings, a former military test pilot and aviation consultant, published this excellent video on the nature of Vmc and what it means to the design and operation of aircraft.