Learn Arabic (Modern Standard): أجواء المطعم هادئة جدا.

In Arabic, non-human plurals are grammatically treated as feminine singular. Since أجواء is the plural of جو (atmosphere) and is non-human, any adjective modifying it must be feminine singular, which is why we use هادئة.

I usually learn جو as 'weather'. Why is the plural أجواء used here to mean 'ambience'?

While the singular جو often means 'weather' or the literal atmosphere, the plural أجواء is commonly used in Modern Standard Arabic to describe the general mood, vibe, or 'ambience' of a physical place or social event.

Why doesn't أجواء have the definite article ال (al-), since the English translation says 'The restaurant's...'?

The phrase أجواء المطعم is an Idafa (a possessive construction). In Arabic, the first word of an Idafa never takes the definite article ال, but it is still understood as definite because the second word, المطعم, is definite.

If I used the singular جو المطعم instead, how would the rest of the sentence change?

If you used the singular, the sentence would be جو المطعم هادئ جدا. Because the singular جو is masculine, the adjective would drop the feminine taa marbuuta and become the masculine هادئ. Both versions are correct, but using the plural is a very natural way to describe an encompassing vibe.