Learn Polish: Moim marzeniem jest zostać fotografem. - My dream is to become a photographer.

The verb zostać (to become) requires the noun that follows it to be in the Instrumental case. The base word for photographer is fotograf, and adding the ending -em makes it Instrumental. This is the standard rule for professions and roles after zostać.

Why is 'my dream' translated as moim marzeniem instead of the basic form moje marzenie?

It is also in the Instrumental case! In Polish, when you use the verb być (here as jest) to say 'A is B', the noun that describes the subject takes the Instrumental case. You can think of the sentence as '[To become a photographer] is [my dream]'. Because 'my dream' is the description here, moje marzenie changes to moim marzeniem.

Doesn't zostać mean 'to stay' or 'to remain'?

It means both! Zostać is a versatile perfective verb. When it is followed by a location, like w domu (at home), it means 'to stay'. When it is followed by a noun in the Instrumental case, like a profession, it means 'to become'. Context tells you which meaning is intended.

Could I change the word order and say Zostać fotografem jest moim marzeniem?

Yes, you absolutely can, and it means the exact same thing. Polish word order is very flexible. However, starting the sentence with Moim marzeniem jest... places the emphasis on your dream, which is a very natural and common way to express your goals in Polish.