You can't win any King v King+pawn endgame if the weaker side takes the opposition
checkmating with pawns on the b- or g- file

By the time you're in the position shown, it's stalemate. Perhaps you could show a position from earlier in the game with knight pawns, from which you think you could/should win, and we can be of more assistance.

Hey, thanks for all your help, guys! Drawing upon your comments, as well as a couple other resources, I figured out a helpful rule of thumb: you want to reach the position below with WHITE to move.
So, here's the odd thing: you have to LOSE the opposition once the kings get to the 8th and 6th ranks, and then bring your pawn up behind your king. (Or, you can have the opposition and just take an odd number of pawn moves so that it's lost when you come to this position)
Play continues Ka6 and then if ...Ka8 you bring your pawn up and black is squeezed out by bringing the pawn up. If ...Kc7 or any other move black gives up the promotion square.
Victory for white :)
No, that's not true. You want to reach it with Black to move. The only time it doesn't make a difference is if the King is on the sixth rank.
With White to move:
With Black to move:
With the King on the fifth rank, Opposition is important. For example
With White to move (meaning Black has the Opposition), the game is a draw:
With Black to move (meaning White has the Opposition), White wins:
Note that if the pawn is on the a or h file, the game is drawn, regardless of Opposition or what rank the King is on. Black shuttles from a1 to b1, and White can't make progress. For example:
I've been reading up on king vs king-and-pawn endgames. I've gotten the concept of opposition down to where I can successfully checkmate with a pawn on every file EXCEPT the b- and g- files. I get stalemated every time on those.
I've looked and looked for the secret to this, but nobody seems to deal with this.
Really tired of getting stalemated.... HELP!? :)
Thanks.