Today I successfully checkmated the black king with just a bishop and knight three times in a row against the computer without consulting any tutorials during that time. It's not easy to do but it doesn't really take so long either. For me, what I did was I read about it in several books. (I often find that reading the same material presented in different books really helps me to learn something). After that I tried and failed a bunch. Then I consulted the books and a video again. After that I started trying and failing again. Then I started to notice some particular situations kept cropping up, and I consulted the books again and learned what to do so I could progress to the point where all that is left is like a regular tactic.
There are really some pretty crucial things. The first is just that you will ultimately have to get the opposing King to a corner that's the same color as the bishop.
Pandolfini's workup on this is pretty good because it breaks it into chunks so first he teaches you how to checkmate with the bishop and knight when the king is already trapped in the correct corner. Then he tries to teach you how to drive the king from the wrong corner to the right corner.
So basically when I was trying it, I would set up the white king, knight, and bishop spaced out on the first rank. One thing I did that makes it a bit more complicated but makes sure that you have the technique rather than just some specific moves is to switch up which color bishop you are using. Then the black king is on some random square in the middle of the board.
It's quite tough to learn how to drive the king into a corner but after doing it a bunch that part starts to get worked out. Once you get the enemy king into the corner, you have to force him along the side of the board to the correct corner. This is tough but especially for the first part if you try it a bunch you will see some things keep happening the same every time and it's easy to start building up the sequence that you will need.
Another tough part is that when you get about halfway along the edge of the board, there are some slightly counter-intuitive moves that let the enemy king come off the edge a little bit, but still keep him trapped. These are the ones I had to learn by rote (although I can perform them going either way). Once you've got that down then it's pretty obvious after a few tries how to make the box smaller and eventually give the mate.
Apparently a lot of people question whether this is worth knowing. For example, Silman in his "Complete Endgame Course" skips it because he says it's not useful as these positions are very rare. There's a couple of reasons that I wanted to learn this. Part of me just feels like it's kind of a mark of distinction that it's not an easy thing to learn, and I feel like it sets me apart just a little bit to have learned it. Secondly, I do believe Jesse Kraai's argument, that learning this isn't so much for the fact of using it in a tournament game and being able to score that extra 1/2 point on the quite rare situation it might come up. The point is it helps you learn to use the pieces together in a simple setting, and that's something that translates to other situations for sure.
On the other hand, I don't feel that I have mastered it. In that I feel I kind of have it memorized right now, but that I could lose it. One possibility for me would be to continue practicing it until it becomes sort of second nature to be able to do it right.
This probably won't translate into an apparent strength gain but I am happy to have accomplished it anyway.