Display Names is like a subscription management tool.
What if your signup wants to get your monthly newsletters, but your weekly ones are too much for them? By using different subscriber-facing names for different lists in your account, Display Names lets your users unsubscribe from one but not the other.
If a contact belongs to more than one list in your account, when they click the Unsubscribe button, they will see the choice to remain subscribed to individual lists.
It is sort of like a last minute check, to allow folks to stay on other lists, if they've decided they want to unsubscribe. As long as they are members of more than one lists that you have Display Names activated for, your unsubscribers can choose which list to stay on.