Learn Polish: Mój dziadek niedługo zostanie emerytem. - My grandfather will soon become a pensioner.

The verb zostać (to become) requires the noun that follows it to take the Instrumental case. Since emeryt is a regular masculine noun, we add the -em ending to make it emerytem.

The translation says "will soon become", but zostanie is a single word. How does the future tense work here?

Zostać is a perfective verb, which means it describes a completed action. In Polish, when you conjugate a perfective verb using standard present-tense endings, it automatically creates the future tense. So zostanie literally means "he/she/it will become".

Doesn't the verb zostać usually mean "to stay" or "to remain"?

Yes, it means both "to stay" and "to become"! You can tell which meaning is intended by looking at what follows it. When it means "to become", it is almost always followed by a noun in the Instrumental case, just like emerytem in this sentence.

How would this sentence change if it was about my grandmother?

The word for grandmother is babcia, and a female pensioner is emerytka. In the Instrumental case, feminine nouns ending in -a change to -ą. So the sentence would be: Moja babcia niedługo zostanie emerytką.