The art of framing the baseball gives the illusion to the umpire that a ball just off the plate actually crossed the plate. It also gives the impression that the ball 5 or 6 inches off the plate just missed. The umpire may get the impression that the pitcher has very good control which can influence his calling of balls and strikes.
When you frame the pitch, try to give the umpire a clear view of where you caught the ball. To do this, catch the ball out in front of your body with a slightly bent elbow. If you catch the ball close to your body, it doesn't matter what you do with your glove, the umpire will not see it. Once you have caught the ball, you move your arm and/or glove a couple of inches towards the center of the plate.
If you call for a pitch on the outside part of the plate, make sure you setup behind the plate a few inches in that direction. This will allow you to give a target on the outer half of the plate without having to reach. This puts you in a good position to frame a pitch.
When receiving the ball and framing it's important to move your body along with your glove. By swaying slightly with your body, you reduce the amount of arm movement required to catch the ball. If you setup on the outer half of the plate and can sway a couple of more inches, you can give the impression that the pitch just off the plate was thrown right on target.
This is a pitch that you want to convince the umpire is a strike. If you have your glove in a vertical position, reach slightly to your left and as you catch the ball, turn your wrist toward the center of the plate. If you catch the ball on the outside portion of your glove, this turn of the wrist can give the umpire the impression that the pitch caught the corner. Hold the glove there for a second to give him a good look. A common mistake of young catchers is to move your entire arm towards the center of the plate. Remember you're trying to create an illusion, not trying to trick the umpire. I trick may work once or twice, but creating a good illusion, may give you an inch or two extra for the entire game.
On a pitch more than a couple inches inside, frame the pitch with your arm and your body. As the pitch comes in, sway your upper body towards the pitch. This makes the pitch appear to be closer to the plate than if you simply stick your arm out to catch it. This technique can save a strike from being called a ball if you happen to be set up outside and the pitch comes across the inside corner.
The picture above shows 2 examples to demonstrate the visual difference that can be made by setup and swaying.
In example 1, the catcher is setup in the center of the plate and doesn't sway to the pitch but reaches only with his arm. Once the ball is caught, the attempt to frame still leaves the glove off the plate, with a lot of arm movement.
In example 2, the catcher has setup a couple of inches to the inside. As the pitch comes in he has swayed his upper body toward the pitch. Even though he still has to reach for the pitch, the movement is not as drastic as in the first example. After catching the pitch the catcher has started to sway his body back. This and a turn of the wrist allows him to frame a pitch that is a few inches of the plate over the inside corner of the plate. The movement is smooth and not drastic. Even if he doesn't get that call, he has given the impression that a pitch 4 or 5 inches off the plate, just missed. This will increase the chance of getting the call on the ball that is an inch or 2 off the plate.
Don't save your sway technique for pitches that are well of the plate. Remember you are trying to give the impression that the pitcher has good control. The less you have to move your glove, the better the impression. With that said, don't go overboard. Staying balanced and comfortable behind the plate is essential.