When all else fails use fire
The Moon children are one of the many mysteries or Majora's Mask. They appear in the moon, on a field with a single tree in the center. Four of them wear the boss remains upon Link's arrival, and they are all running around the tree, while the 5th wearing Majora's Mask is sitting by themselves under the tree.
By playing a game of "hide and seek" with each child, and giving them your non transformation masks, you can receive the fierce deity's mask from the child wearing Majora's Mask. Also worth noting is how they react to being given a transformation mask,
"Not that! Hurry...put it away..."
Now when you complete these games, the children ask for more masks, and then ask Link a question
Odolwa Child: Your friends... What kind of... people are they? I wonder... Do these people... think of you... as a friend?
Now this seems like a pretty straight forward question. It kind of makes me think of the Deku Princess, and the monkeys, or more likely the skull kid, and the giants.
Goht Child: What makes you happy? I wonder...what makes you happy...does it make...others happy, too?
This also seems pretty straight forward as well as far as I can tell. Also it can be attributed to the skull kid, as he caused mischief for the people of Termina, and now Majora's Mask, or the moon child, is pondering about it's actions.
Gyorg Child: The right thing...what is it? I wonder...if you do the right thing...does it really make...everybody...happy?
Now this is what stuck out to me at first. If you do the right thing, does it really make everybody happy?
This immediately makes me think of Mikau. He did what appeared to be the right thing, and tried to save Lulu's eggs, but ended up dying. This can also be attributed to Link who posses as Mikau to keep the band happy.
Twinmold Child: Your true face... What kind of... face is it? I wonder... The face under the mask... Is that... your true face?
This is also an interesting quote. The child refers to Link's face as a mask. Now this can go a number of ways, from how we put on a metaphorical mask when we go out into the world, or maybe Link is asking this question about the skull kid, and the child was just saying what Link was thinking.
Majora Child: ...Everyone has gone away, haven't they? Will you play...with me? You don't have any masks left, do you? Well, let's do something else. Let's play good guys against bad guys... Yes. Let's play that. Are you ready? You're the bad guy, and when you're bad, you just run. That's fine right?
Why these children are asking these questions is also a bit of a mystery. Some say it's to make Link question himself, and maybe weaken his mind, or maybe it's Majora's mask wondering about these things. After being attached to skull kid for a time, Skull kid may have imprinted his memories, and thoughts onto the mask, and it's now curious about things such as what makes one happy vs. what makes others happy.
The Moon children also resemble the Happy Mask Salesman, who has a similar fondness for masks, and making people happy. It could be they are all of the same race.