[fen "3r4/6R1/1kp5/8/8/8/5K2/8 w - - 1 61"]
1. Ke3 Kc5 2. Rg4 Kb5 3. Ke2 c5 4. Rg3 Kb4 5. Ke1 c4 6. Rg4 Kb3 7. Rg7 c3 8. Rb7+ Kc2 9. Rb4 Re8+ 10. Kf2 Kd2 11. Rd4+ Kc1 12. Rc4 c2 13. Rc7 Rb8 14. Ke2 Kb2 0-1
This is a situation where the king is cut off by one file from the pawn. The king has to make a timely jump to the second rank and prepare to exchange rooks. In the meantime, the king shall remain on the third or fourth rank.
When is the jump "timely"? In a king and pawn vs. king endgame, pawn on the fifth rank, the key squares the king needs to reach in order to promote the pawn are b3, c3, and d3 (in this case): no matter whose move it is. White will threaten to exchange rooks when it is clear the black king cannot reach these squares because the white king will be shouldering, i.e. in the way.
It is clear that 3. Ke2? was a losing move. The rook isn't even ready to be traded, so right away this move is premature. Instead assume that the rook was already on the back rank (Rg1 instead of Rg7). 3... c5 4. Rd1 Rxd1 5. Kxd1 Kb5 6. Kc1! And it's clear that the black king won't access b3, c3, or d3, because white wins the opposition: draw.
Now let's assume that white stuck with his plan of exchanging rooks: 4. Rg1. 4... Kb4 5. Rd1 Rxd1 6. Rxd1 Kb3 and the ending is won for black. Note the hasty 6... c4?? loses to 7. Kc2.
As for the rook, frontal checks with distant effectiveness must be performed (that is, checks with your rook on the back rank, far away from the enemy pawn). The defending rook must not leave this back rank.
While the rook isn't delivering checks, he should be threatening to capture the pawn: Rc1 is the correct position. The correct maneuver is to check the king, separating him and the pawn by a file (Rb1+, Rc1; but not Rb1+, Ra1+? too many checks) -OR- causing him to hide behind the pawn (Rb1+, Rc1 - here he cannot advance the pawn, so you move the king upward with Ke5 and threaten to capture on c6). In both cases you threaten to capture the pawn, so he must use tempi in defending the pawn by moving the rook -OR- moving the king, in which the latter leads to perpetual check. Draw.
White could have troubled you with 4. Rg1 (threatening Rb1+ - though due to his misplaced king, still loses after 4... Rd5! Now 5. Ke4 isn't possible) but instead he's escorting you to his back rank: Slowly, but surely with 4. Rg3? which does, to put it bluntly, absolutely nothing.
Of course, the Philidor position is still reachable because the pawn is so far from promotion i.e. it has not reached the fifth rank yet.
So how should have this game proceeded?
1. Rg1 preparing to restrain the pawn from sole advancement. Now 1... Kc5 doesn't make any sense because of 2. Rc1+, where 2... Kd6? is a clear draw, and 2... Kb5 3. Rb1+ Ka5 4. Rc1 sees no progress as well.
Black can try 1... c5, but this now allows 2. Ke3! Kb5 3. Rb1+ Ka4 4. Rc1 Rd5 5. Ke4!! Now the king, not ill-positioned as before, springs into action.
For further reading, please see Ending 59 of Jesus De La Villa's 100 Endgames You Must Know.