Believe it or not, the answer to this is pretty simple: volume. Even a small arena is pretty large, and when it's filled with raucous fans, it becomes very loud. The pipe organ (and later, the electric organ) was able to be heard over that din, and thus lend a "psychological accompaniment" to whatever event it was being used for.

When the old Chicago Stadium opened in 1929, it had a massive pipe organ installed, becoming the first arena in the US, if not the world, to have such a feature. Some 3663 pipes made up the beast which was said to be as loud as 25 brass bands. As an aside, when the Chicago Stadium was torn down, the organ was sold and removed, and eventually wound up in a warehouse in Arizona. There, it was destroyed in a fire.

Of course, enormous pipe organs were tremendously expensive. The rise of the electric organ made it possible to have similar volume for a fraction of the cost. While long-time fans rue the loss of the old pipe organ sound, it only makes sense financially to use the very similar sounding Hammond instruments.

Interestingly, the first use of an organ in baseball also occurred in Chicago, in 1941 at a Cubs game in Wrigley Field.

source: "From Town Hall to Play Ball!: The Origins of the Baseball Organ."