Learn Polish: Nie rozumiem, dlaczego sześcioro dzieci płacze. - I do not understand why six children are crying.

Sześcioro is a collective numeral. In Polish, you must use collective numerals instead of standard numbers with certain types of nouns. The most common ones are neuter nouns that end in -ę in the singular, like dziecko (child) or szczenię (puppy), and nouns that only exist in the plural, like drzwi (doors). You also use them for mixed-gender groups of people.

What case is dzieci in this sentence?

It is in the Genitive plural. When a collective numeral acts as the subject of a sentence (which is the Nominative case slot), it forces the noun that comes after it into the Genitive case.

There are six children, so why is the verb płacze (singular) instead of płaczą (plural)?

This is one of the trickiest rules for Polish numerals! When a collective numeral (like sześcioro) is the subject of a sentence, the verb must be in the 3rd-person singular form. You can think of the grammar as saying "A group-of-six of children is crying."

I notice a comma before dlaczego. Is that required?