How to win 3 pawns and a rook vs 3 pawns and a rook endgame

You can't win unless he makes a mistake. You allowed his rook to just take your pawn. It was still a draw, but all you needed to do is to keep your king near your pawns so that he can't just take. Your rook on e3 was restricting his king. So if your king didn't just leave the defense of the pawns you wouldn't have lost. 

Even then it was a draw, but you just let that pawn go, you marched further away with your king from where the action was happening.

You march your king up the board to attack enemy pawns or to support your own pawns to the promotion line. Your king marched to nowhere.

One of the skills of GM-play is knowing when to wait because there is nothing to do. This is one of those places. Just defend your own pawns and invite your opponent to take undue risks.

Thanks for the comments ! Both of your points are really instructive! Indeed, I was desperately moving my king away from his duty to try to breakthrough. I should have played solidly and  be patience. 

No general principles and it may be already lost by move 54, but doesn't 54...Re2+ just give away your last pawn if he'd played Kf3?

Edit: I've cheated and looked it up on Syzygy and it's actually a tablebase draw before that move and in fact it's still a draw after his response.

You lost because you pressed to find a win in a position where there were no realistic winning chances at all. It's great to want to win,  but you can't win in such a dry position simply by force of will.  Your aggressive attitude will be an asset to you, but you have to temper it with realism. Caruana and Carlsen would probably shaken hands after say, 42.Kh2

Yeah, this is something about endgames that it took me a while to figure out.  In the opening and middlegame -- and often in the endgame too -- it pays off to play as actively as you can.  But there times in an endgame where there's nothing to be accomplished through active play, and all you can do is shuffle pieces around, keeping them decently positioned for the defense.  This is one of those endgames, and when you find yourself into those, the surest way to lose is through trying too hard to win.

Thank you all. This time I was really patient in the end game as you guys suggested, and I won because my opponent was desperated to win and blundered the king.

At the  he did blunder a king. The reason was He would most likely end up in a draw so in his desperation to win he must have tried something else to hopefully win. Turns out he didnt think before he acted