Why does a pilot bank up to 5 degrees into the operating engine following failure of the other 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.