What is the idea here?

Apparently, the correct move is Rg6, and engines agree, but I don't see why this is clearly better than Ng6. Can anyone offer an explanation?

I suppose that makes sense: doubled rooks on an open file seems attractive enough. I was expecting a tactical move, since the place I'm getting it from says this position should test tactical skill.

There is a lot of finagling that can be done with a rook threatening the pawns and dominating the open file.

I think the use of "correct" may be a bit facile here.  White's endgame is awfully good and there are doubtless many roads to victory in such a situation.

Moreover White's knight has extra mobility than the rooks. He can go to g4 to pressurize Black or go wild in the queen side if Black succeeds to lock the position on the king side.

Also if Rxg6 then hxg6 or fxg6 gets a supported passed pawn.

Ng6 doesn't do anything. Black plays  Rh7, and white has no follow up, while Rg6 threatens to take on f6 and at the same time prepares for white to double the rooks on g-file. I would assess the position a moderate advantage for white.

I would assess it as huge advantage to White.

Don't follow a "??" by a "!"

This problem was originally from a suite of problems called BT2450, which is meant to determine the tactical Elo rating of a chess engine at a glance. A friend of mine who had created such an engine could not figure out why his engine could not find Rg6, which other engines found this move extremely quickly (with depth 4-6). It turns out he hadn't put enough emphasis on doubled rooks on open files, which is now a much bigger bonus.

Thanks for all the help.

Ah, that's an interesting background! 

fxg6 is clearly better, after nf5 black is completely paralyzed

The difference is to trade pieces, else White doubles on g-file. The text plans to play Ng6, but leaving the option open to capture with a pawn to make a passed pawn. It also makes the win easier by trading off pieces.

After hxg6 black can block with his king or bishop on g7 nf5 eliminates this, but im sure that would win also

Besides which, White also has the queenside to play around on.

hiya peeps