Learn Russian: Я не пью из грязных чашек.

The preposition из (from/out of) always requires the Genitive case. Since we are talking about plural cups, we need the Genitive plural. The dictionary form of the adjective is грязный (dirty). Because its stem ends in a hard consonant, it takes the standard Genitive plural ending -ых.

The dictionary word for cup is чашка. Why does its Genitive plural form, чашек, have an extra е in it?

To form the Genitive plural of most feminine nouns ending in -а, we simply drop the -а. However, dropping it from чашка leaves an awkward double consonant at the end: чашк. To make it easier to pronounce, Russian inserts a "fleeting vowel" between those consonants. In this case, an е is inserted, resulting in чашек.

There are a few Russian words for "from" (like от and с). Why do we use из in this sentence?

You use из when an action involves taking something from the inside of an object or space. Because you are taking liquid out from the inside of the cup, из is the correct preposition.

I know the verb for "to drink" is пить. Why does пью have a soft sign (ь) in it?