Peter thins them out

This would be a idiom, to me. It is a phrase that does not have specific syntactical meaning, but only has the meaning it has as a complete phrase. The same would be true of phrases such as "grosses me out," "turn it on," an so forth.

You could make the case, I suppose, that "turn it on" derives from "Turn the switch to the 'on' position." In the same way, "thin them out" might derive from the idea, "thin the flock by taking some out."

Nonetheless, that actual phrase has no meaning except as a complete phrase.

There is a shorter version, “thin out.” It is the same, though: "Bob, thin out the flock before you quit for the day."

(I have no authority on this, aside from understanding English well.)

I visited the link of the Japanese program. As I don't speak Japanese, most of it was beyond me, but I did glean that it's a game show of some sort, not a news or educational program. Also, it seems that the show is more concerned with sensationalism than accurate reporting. Note, the program shows a quote from the book:

...Peter thins them out;
---------------------------------
a grownup dies; and Peter was killing them off vindictively as fast as possible.
Then having given the necessary instructions to the Redskins...

This is extremely dishonest and misleading, and demonstrates that the program's producers are only interested in sensational headlines, not actually informing their viewers. The quote is actually two different quotes from two different sections of the book, combined and taken out of context. In particular, the second part "a grownup dies; and Peter was killing them off...", in the book, is clearly referring to Peter's distaste for adults and has nothing to do with killing off his band of aging kids (or anyone else for that matter). The full quote is:

"...he was so full of wrath against grown-ups, who, as usual, were spoiling everything, that as soon as he got inside his tree he breathed intentionally quick short breaths at the rate of about five to a second. He did this because there is a saying in the Neverland that, every time you breathe, a grown-up dies; and Peter was killing them off vindictively as fast as possible."

Peter isn't actually killing anyone. He's imagining doing so simply by breathing in and out quickly.

To address your original question, "thins them out" clearly means reducing their ranks like thinning out a herd of animals, but metaphorically, this could mean anything. It could just as likely refer to them being killed by pirates, or leaving Neverland upon growing up, or being banished or shunned, etc. While it does sound ominous and a bit disturbing, it's vague enough to leave the details to the reader's imagination (typical Barrie, yes?). While it's been a while since I've read it, I don't think there's anything else in the book that truly suggests that Peter is doing in his cohorts. In fact, elsewhere in the book or in later works, aren't they revisited as adults?

Yes, I agree it's a metaphor, child's play or imagination is meant, IMO :)

My mother always said that when the boys grew up, Peter made them leave the island and go back to the real world and that's what thins them out meant. Was she just being kind to my child's mind???

Isn't it left to the child reader's fantasy exactly what's meant? Peter could be making them jump off a cliff when they get too old -- or returning them to the world -- or just drumming them out of the ranks and they're on their own. However he reduces the ranks . . . and a child who is reading Peter Pan is not shying away from fantasized aggression, with the safety net of its being just a story. As the child's development requires at the moment. Peter is a bloody-minded ruthless hero who is both an eternal child and the equal and nemesis of one hairy bad grownup. What could be more glorious?

About things like "turn on" or "thin out" -- these aren't really idiomatic phrases, they are compound verbs that we inherited from the germanic roots of the language. What looks like a preposition is really an adverbial prefix; some of them detach from the stem verb, some of them can't, and some can go both ways, and the meaning is distinct depending on which way it goes. Example: the difference in meaning(s) between "uphold" and "hold up." You could conceivably "hold up" the founding principles of a nation, although "uphold" would be better; but you really could not "uphold" a liquor store with a gun.

"The boys on the island vary, of course, in numbers, according as they get killed and so on; and when they seem to be growing up, which is against the rules, Peter thins them out..."

This is a slightly larger quote and I can't say whether Peter does kill them but I thought that quote as a whole made it seem more of a possible idea. (Also he is quite violent and just for the fun of it too- killing pirates and wreaking havoc in the midst of battles are a couple of examples)
I just wanted to say I wouldn't expel the idea, it's certainly an interesting thought to contemplate