Learn Hardcore Russian: На платформе я читаю журнал, пока жду поезд. - On the platform, I read a magazine while I wait for the train.

Because на платформе uses the prepositional case to mean location: (being) on the platform.
На платформу would be accusative and would mean motion/direction: (going) onto the platform (e.g., Я иду на платформу).

What case is платформе and how is it formed?

Платформе is prepositional singular of платформа.
Many feminine nouns ending in -а take -е in the prepositional: платформа → (на) платформе.

Why does Russian use на (on) for a platform instead of something like в (in)?

Russian treats many “surface/area” places as something you’re on, not in: на платформе, на остановке, на площади, на улице.
В is used more for enclosed spaces: в вокзале (in the station building), в метро (in the метро system / subway).

Is я necessary here? Can it be dropped?