Learn Farsi: برگ‌ها روی شیشه‌های ماشین ریخت.

This is the main rule for inanimate plural subjects! In Farsi, when the subject is plural but not alive (like leaves, books, or cars), the verb is usually singular. It is as if the leaves are being treated as one big collective group rather than individual actors.

Could I use the plural verb ریختند instead?

You could, but it would sound very poetic, like you are personifying the leaves and giving them human-like qualities. In everyday spoken and written Farsi, native speakers stick to the singular verb ریخت for things that are not alive.

The translation says fell, but does not ریختن mean to pour or to spill?

Yes, but it is a very flexible verb! While it means pouring a liquid, it is also the standard verb for lots of small things falling or scattering, like leaves falling from trees, hair falling out, or crumbs spilling. If you were talking about a single heavy object dropping, you would use the verb افتاد instead.

How are the words connected together here?

Through the ezafe! You need an ezafe to connect روی (on) to شیشه‌های (windows), pronounced ru-ye. Then you need another one to connect شیشه‌های to ماشین (car). Because the plural word شیشه‌ها ends in a vowel, the ezafe is written with a ی and pronounced as ye. The whole phrase sounds like: ru-ye shishe-hā-ye māshin.