We talk so much about receiving the pitch as a catcher, it seems maybe the position should be called “Receiver” instead of Catcher. Regardless, receiving is the most important skill a catcher must learn. Blocking and throwing are also critical skills for a catcher to learn and master but, those skills are called upon less frequently in a game whereas a catcher has to be prepared to receive every pitch.
There are two schools of thought around how to properly receive a pitch. One heavily focuses on truly framing a pitch around the strike zone. In this approach the catcher is taught to rotate the glove upon receipt of the pitch to literally create a “frame-like” view of the pitch on the border of the strike zone. Refer to the example below how the glove is turned vertically (thumb up) for glove-side pitch (inside to RH batter). Similarly, this approach teaches that the glove should be turned thumb down to frame the borderline arm side (throwing arm) pitch.
The other school of thought around receiving is to present as consistent a glove position regardless of where the pitch is received. This, combined with the technique of beating the pitch to the spot and funneling toward the strike zone will result in far more strikes being called on those “stri-balls” (pitches that are borderline strikes or just off the edge which a good catcher can manipulate to appear a strike). In the image below you can see Buster Posey receiving a glove-side pitch (his body lean to glove side indicates the pitch is inside to a RH batter) with a horizontal glove. Catchers are taught to set-up with the glove in this horizontal position (thumb parallel to the ground).
Now let’s take a look from an umpires perspective. In the two images below (credit to @dnacatching) you will see the view from the umpires perspective. The first image show you the vertical glove position and the second shows you the horizontal glove. It is clear to see the horizontal glove presents more of the glove over the plate providing the umpire the perspective that the pitch is clearly a strike.
If you notice, the wrist/forearm are actually closer to the plate in the vertical glove position yet less of the glove and where the ball is in the glove is actually over the plate. With the horizontal glove position the forearm is further from the plate but the glove is covering far more of the plate. Additionally, where the ball is in the glove is clearly over the plate.
Another perspective you have to consider is the presentation of the glove from pitch to pitch. If the catcher only turns the glove vertically for borderline pitches and pitches well outside the zone while maintaining a horizontal glove for pitches clearly in the strike zone it actually gives the umpire the impression that you are trying to steal a strike and he is less likely to call it a strike because of that. By presenting a horizontal glove position as frequently as possible there is less difference between each pitch around the zone so the umpire see consistency and is more likely to give you a strike call.
Below are a couple video examples.
You’ll notice in the first clip of Yadier Molina that the pitch is low and inside. He actually turns the glove vertically (thumb down) to catch the pitch but then turns the glove quickly back upward into the zone to a horizontal position. The point is, he doesn’t attempt to frame it with a vertical glove position
Below you will see Yasmani Grandal receiving an glove-side pitch (inside to a RH batter) and he maintains a horizontal glove position to secure a strike.