Idea #1: According to Kopaev, "When the [defending] king is cut off two or more files from the pawn, [the attacker] always wins." White to move with the rook on a1, a2, a3, a4, or a5 is a trivial win. For example, 1.Rg3+! Kh7 and White wins per the above rule.
Idea #2:
Black's king is ideally placed on g7. From there, he controls the f6, f7, and f8 squares, so White's king can't seek shelter from rook checks by fleeing to the kingside. Also, his king is only two moves away from the queening square.
Black's rook is on the best square possible on the b-file. Of course, being on a8 would be an improvement over b8, but that's just not possible. On b8 it prevents the White pieces from advancing to the eighth rank.
Idea #3 - White now moves his rook to a1, preparing to support his passed pawn or cut-off Black's king. Black does the only natural thing in this position... he checks!
Unfortunately his checks are doomed to failure. There are only 2 squares between his rook and the pawn. A successful defense of this sort requires 3 squares... aka checking distance. White capitalizes on Black's misfortune in a typical manner.
Idea #4: The final decision for White is whether to use his rook to (a) cut-off Black's king or (b) support his pawn. If you calculate just a couple moves ahead, you'll see the cut-off approach goes nowhere. The support approach yields fantastic results!
As I said in the beginning, this puzzle is moderately difficult, but it's also very instructive. I think better endgame results are in store for the brave few who attempt to unlock its secrets! I hope you enjoyed. :)
Wow, you made it through! Between Barbier,Saavedra 1895 and Kopaev, 1953, I don't understand why some people call endgames dull. :)
I believe no human nor computer has ever fully mastered the endgame. And there's always one endgame just a little more complex than the one you know! 
For me, the above ending is completely boring. I imagine two grandmasters would shake hands and walk away, if they respected each other.
But they wouldn't in the above position! The White player's Kalashnikov and the Black player's Karpov. White almost wins, but he makes a small mistake ten moves later, and Black is able to secure the draw six moves after that. I imagine neither player was feeling bored about the position. And it was great fun to watch!
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This position was composed by Kopaev in 1953. It's medium difficulty, but the solution is instructive and it's a good position for a tournament player to know. If you want a hint, I covered a couple related positions in Not Quite Lucena III. Enjoy!!