Rook vs knight end game, how to win?

"you can step into a fork"

True, but there is no malevolent force that causes forks to appear out of nowhere. Spend 15 minutes learning about the chess board. Put a knight on various squares on an empty board and visualize the squares it can fork in one move. Become conscious of the fact that two pieces (eg. king and rook) on the same-colored squares may be forkable. Remember that a knight always moves from a white square to a black square and vice versa. So if your king is on a black square and the enemy knight is on a light square, there is no way he can move and give check. All this is elementary, it's like learning your elementary multiplication tables. And the minutes you spend mastering them will take a lot of the mystery and fear out of the game 

I think you just need to learn endgames. It's hard to explain everything to you that a stronger player takes for granted. With white you have an edge because a rook is just better than a knight. The rook can get to any square on the board in just two moves with no obstacles, the knight takes three moves just to get to a square right next to it. The rook in these situations can almost always trade itself for the knight to convert to a winning pawn endgame, you just have to know when the pawn endgame is winning but to know that you would have to know pawn endgames. In the diagram you posted white can force just such an ending with Rg4 and taking on g6 but would it be winning/drawing/losing? Why would it be winning/drawing/losing? If you can't answer that then you have a gap in your chess knowledge that needs to be filled before you delve too deeply into this endgame. The simple idea in a rook vs minor piece endgame is to either have your rook on the 7th rank with his king on the 8th rank and your king on the 6th rank or to get your rook behind his pawns, if he has to keep advancing them to save them eventually he won't be able to save them. So in your game, the easiest way to win is probably Re8 and swing behind the queenside pawns.

That would at best be drawing for white with a high risk of losing.

And one move thing: How did you lose that game? I can see not being able to win, but losing that game took something special.  Stockfish evaluates the diagrammed position at +8.54. I assume you lost your rook for nothing?

I gave it a shot.

I set the position up on my Fidelity Elite Avant Garde 2265, Level 5, though I'm very rusty with actual pieces.  But, I only made one mistake, which was such an obvious blunder that I discounted it with a quick take-back-move, and tried something else instead.

Others are correct, and it was my approach, too.  Get your Rook to Black's back rank and get rid of a pawn over on the left, somehow.  Attack them from behind.

Your Rook's huge mobility over the Knight is your key to success.  At one point I made a 2nd blunder with my Rook, where the Knight could capture it by forking my King, but I let the Knight capture the Rook, and then took the Knight, doubling my pawns.  It didn't matter.  My King finally took the right-side pawn, and Black's King couldn't cover all of my pawns without my King returning to the defense of at least one of them.

The thing about a lone King is that He has to move down in tandem with multiple pawns.  Capturing one will allow another to advance out of reach.