Learn Polish: Mój szef zwykle jeździ do centrum. - My boss usually drives to the center.

Because of the word zwykle (usually). Jeździ is a habitual motion verb, used when someone goes by vehicle regularly or in multiple directions. If he were driving there right now on a single, specific trip, you would use the unidirectional verb jedzie.

Does jeździ mean he is the one driving the car?

Not necessarily. Jeździć just means to go by vehicle. He could be driving his own car, but he could also be taking a bus, a tram, or a taxi. If you want to specifically say that he is the one steering a car, you would use a different verb: prowadzić.

Doesn't the preposition do require the Genitive case? Why hasn't the ending of centrum changed?

You are right that do (to) always requires the Genitive case. However, centrum is a foreign loanword in Polish. Nouns ending in -um are neuter and do not change their endings at all in the singular. So it stays centrum no matter the grammatical case!

Is szef only used for a male boss?

Yes, szef is a male boss, which is why it uses the masculine word for my (mój). If your boss is a woman, you would use the feminine form szefowa and say Moja szefowa zwykle jeździ do centrum.