As best as I can parse it, it is the interaction between the Light weapon property, the Nick Weapon Mastery, and the Dual Wielder feat:
Assume you are fighting with a dagger and shortsword, and take the
attack action to make an attack with the dagger.
Light(1) property allows you to make an extra attack with the shortsword as a
bonus action.
So
Dagger, then
Shortsword.
Nick (2) turns that shortsword attack (from the Light property) into part of the same
attack action as the dagger attack.
So
Dagger, then
Shortsword.
Dual Wielder(3) allows you to make an extra attack with one of your weapons as a
bonus action. (We'll assume shortsword, though it could be either.)
So
Dagger, then
shortsword, then
shortsword.
Crucially, while Dual Wielder requires the Light property to grant its extra attack, this extra attack is from the Dual Wielder feat, not the Light property, allowing both to be used for two additional attacks.
Without Nick however, both of these extra attacks would take your bonus action, so you could only make one.
Dual wielder also allows your bonus action attack to be with a non-Light weapon. However the extra attack due to the Light property requires the other weapon be Light. (I'm not entertaining weapon-swapping shenanigans for this.)
This is RAW to the best of my understanding, and I believe it is also RAI from comments made by Jeremy Crawford when he was cornered and asked by a youtube creator.
(1) When you take the
Attack action on your turn and attack with a Light weapon, you can make one extra attack as a Bonus Action later on the same turn. That extra attack must be made with a different Light weapon, and you don’t add your ability modifier to the extra attack’s damage unless that modifier is negative. For example, you can attack with a
Shortsword in one hand and a
Dagger in the other using the
Attack action and a Bonus Action, but you don’t add your Strength or Dexterity modifier to the damage roll of the Bonus Action unless that modifier is negative.
(2) When you make the extra attack of the
Light property, you can make it as part of the
Attack action instead of as a Bonus Action. You can make this extra attack only once per turn.
(3) When you take the
Attack action on your turn and attack with a weapon that has the
Light property, you can make one extra attack as a Bonus Action later on the same turn with a different weapon, which must be a Melee weapon that lacks the
Two-Handed property. You don’t add your ability modifier to the extra attack’s damage unless that modifier is negative.