When does white do it? Why does white do it? What kind of strategies should both sides adopt in such games? What kinds of games does it give rise to (i.e. positional or tactical)? Which players are well known practitioners of such a setup as white?
The most common way to end up with queenside castling for white in the KID is via the Saemisch Variation
5.f3. Black often attacks on the kingside in the KID, and the Saemisch is typically a way to avoid that, and ideally to attack on the kingside oneself. Of course, whole books have been written about what's going on in this variation, and I don't play either side of it myself, so this answer only addresses the last part of your question, concerning well-known practitioners.
The link above notes that
The Sämisch has been played by numerous grandmasters, including world champions Mikhail Botvinnik, Mikhail Tal, Tigran Petrosian, Boris Spassky, Anatoly Karpov and Garry Kasparov.
True enough, and so did Alekhine. But if my database is to be trusted, none of these champions has all that many games on record in the Saemisch, and even less against serious competition. For instance, Karpov played such a setup twice in his 1990 world championship match against Kasparov, scoring a draw (round 21) and a win (round 23), which is a nice endorsement. But he seems not to have played that way often; in my database, the only other classical game for Karpov in this setup against a player who is truly a peer of his is a draw against Timman from round 18 of their 1993 FIDE championship match.
The late Svetozar Gligoric has 18 games on the white side of this in my database (scoring 12.5 points), and his games could be particularly enlightening since he was a well-known proponent of the black side of the King's Indian; in fact, I also see him scoring 22.5/37 on the black side of this, so there's plenty of Gligoric games from boths sides to chew on.
But the most frequent assayer that I find is Florin Gheorghiu, and he's also a very successful one, with a phenomenal 35.5/43 score. So if you want a "hero" to follow for this kind of setup, Gheorghiu might fit the bill nicely. (Or conversely, if you want to know what might be coming at you as black.) To whet your appetite, what follow are two of his wins, each with Gligoric playing black in fact. The first is from the 1966 Havana Olympiad:
[FEN ""] [Event "Havana ol (Men) fin-A"] [Site "Havana"] [Date "1966.??.??"] [Round "4"] [White "Gheorghiu, Florin"] [Black "Gligoric, Svetozar"] [Result "1-0"] [ECO "E87"] [EventDate "1966.10.25"] [PlyCount "81"] 1.d4 Nf6 2.c4 g6 3.Nc3 Bg7 4.e4 d6 5.f3 O-O 6.Be3 e5 7.d5 Nh5 8.Qd2 f5 9. O-O-O Nd7 10.Bd3 Ndf6 11.exf5 Bxf5 12.Nge2 Qd7 13.g4 Bxd3 14.Qxd3 e4 15.Nxe4 Nxe4 16.gxh5 Nc5 17.Bxc5 dxc5 18.hxg6 Rae8 19.gxh7+ Kh8 20.Nc3 Bh6+ 21.Kb1 Re3 22.Qg6 Qg7 23.Rhg1 Rfxf3 24.d6 cxd6 25.Rxd6 Rf8 26.Nd1 Ref3 27.Qxg7+ Bxg7 28.Rg5 b6 29.a4 Kxh7 30.a5 Bd4 31.axb6 axb6 32.Rxb6 Re8 33.Rg2 Re4 34.Rc6 Rh3 35.Rd2 Kg7 36.Nc3 Re1+ 37.Kc2 Kf7 38.Nb5 Be3 39.Rd7+ Ke8 40.Rb7 Re2+ 41.Kb1
And the other is from the 1968 Lugano Olympiad:
[FEN ""] [Event "Lugano ol (Men) fin-A"] [Site "Lugano"] [Date "1968.11.03"] [Round "10"] [White "Gheorghiu, Florin"] [Black "Gligoric, Svetozar"] [Result "1-0"] [WhiteElo "2520"] [BlackElo "2575"] [ECO "E87"] [EventDate "1968.10.17"] [PlyCount "95"] 1.d4 Nf6 2.c4 g6 3.Nc3 Bg7 4.e4 d6 5.f3 O-O 6.Be3 e5 7.d5 Nh5 8.Qd2 f5 9. O-O-O Nd7 10.Bd3 Ndf6 11.Nge2 fxe4 12.Nxe4 Nxe4 13.Bxe4 Nf6 14.Bc2 Bf5 15. Ng3 Bxc2 16.Qxc2 c5 17.dxc6 bxc6 18.h4 Qa5 19.Kb1 e4 20.h5 Qe5 21.Nxe4 Nxe4 22.Bd4 Qe7 23.Bxg7 Kxg7 24.h6+ Kf7 25.Rhe1 Rae8 26.Rxe4 Qf6 27.Qa4 Rxe4 28.fxe4 Rc8 29.Qxa7+ Qe7 30.Qd4 Re8 31.Rf1+ Kg8 32.Rf4 g5 33.Rf5 Qxe4+ 34.Qxe4 Rxe4 35.Rxg5+ Kh8 36.Rg7 Re6 37.Rc7 Rxh6 38.Rxc6 Kg7 39.c5 Rh1+ 40.Kc2 dxc5 41.Rxc5 Rg1 42.Rg5+ Kf6 43.Rg8 h5 44.Rf8+ Kg5 45.Rf2 Kg4 46.Re2 Kg3 47.a4 h4 48.a5