How can a load increase affect the line voltage by dropping it on that circuit?
The source as well as the conductors between the source and whatever point you are interested in all have resistance/impedance. More load you apply the more voltage will drop. Larger than needed source or larger than needed conductors just need even more load on them before the results become significant.
If you have a production facility and come take voltage measurements on a "down day" they likely will tend to be higher than on a day with normal production levels, simply because there is less current being drawn and less voltage drop across the source as well as less voltage drop on service and feeder conductors. Individual branch circuits still in operation, (not all items necessarily go down on a down day) might still see some voltage drop that you don't see elsewhere at that time.
So to have 480 to 120 transformer you have a 4 to 1 turns ratio. But lets say on a down day the load is low enough the actual primary volts goes up to 520 then the secondary will increase proportionally to 130.
On top of that say the normal loading brings the incoming 480 down to 465 then the secondary volts will drop to 116.25. But depending on how much load is on that 120 volt side it could drop even further just because of impedance in the secondary and the secondary conductors.
Selecting a tap to use on the primary of said transformer can bring output volts up or down (you usually have 5% change between each possible tap configuration) But keep in mind you should probably select a tap based on where the loading conditions are the most prevalent. Because it will still go up and down as loading conditions change, and is still effected by any change in incoming primary voltage as well. Often in a production situation if the 480 volt loads are decreased significantly so will a lot of the loads on the secondary, simply because production levels changed, so over compensating on tap setting can put you at pretty high voltage on the secondary when overall load levels are low.