Learn Farsi: همه‌ی فامیل‌ها در رستوران جمع می‌شوند. - All the relatives gather in the restaurant.

This is the ezāfe connecting the word "all" to the noun that follows it. Because hame ends in a silent h (which makes an 'e' sound), we add -ye to link it. So hame-ye acts like "all of".

Does the noun following hame-ye always have to be plural?

Yes, when it is a countable noun! If you are saying "all the [things]", the noun needs a plural ending like -hā. This is different from numbers, which take singular nouns. Because "relative" is countable, we must say hame-ye fāmil-hā.

Doesn't fāmil mean "family"? Why is it translated as "relatives" here?

In conversational Farsi, fāmil is most often used to mean an individual extended family member, or a "relative". Therefore, the plural fāmil-hā means "relatives". If you want to talk about a single, nuclear family unit, the word khānevādeh is usually the better choice.

How do the verb parts in jam' mishavand work together?

This is a compound verb made of jam' (gathered or collected) and shodan (to become). mishavand is the present tense form for "they". So literally, the relatives "become gathered" in the restaurant.