Why did the Phantom Troupe let Gon and Killua go? In the very first episode of the Greed Island arc, Gon and Killua meet up with two of the Phantom Troupe members while in the auction.

The members say they will no longer hunt our heroes because... something to do with their leader's Nen surviving after death? I watched the episode again, but it doesn't make any sense.

Why is the Phantom Troupe no longer interested in revenge against Gon et al?
Kurapika placed his judgement chain on the Chrollo Lucifer, the Phantom Troupe's leader. The Phantom troupe knows that and that Lucifer is currently seeking the help of a Nen exorcist in order to remove it.

Now why did they let Gon and Kirua go? Well because they want nothing to do with Kurapika anymore! If the person who placed a Nen dies, his Nen becomes much stronger after death, and a much more powerful Nen exorcist would be required in order to remove it. Actually, only a handful of exorcists have the skill to remove the Nen left by a dead user [1]. Or, since Lucifer is at this point weak because deprived of his Nen, he might even die.

The Phantom Troupe knows this and therefore they do not want to have any more direct or indirect interactions with Kurapika. If they were to harm Gon and Killua, Kurapika would surely come seek vengeance once more. If they were to fight off Kurapika, they would have to hold back to make sure he survives the fight. That would be not be an easy feat as Kurapika is not so weak. 

[1] Chapter 120, page 13 (not shown here for compactness)

Some of this is explained on page 12 of Chapter 120: