Learn Hardcore Polish: Bawełna jest czysta.

A big clue is the ending -a. In Polish, many nouns ending in -a are feminine. Here it is also singular and in the nominative case, because it is the subject of the sentence.

Why is it czysta and not czysty or czyste?

Because Polish adjectives must agree with the noun they describe in gender, number, and case.

Since bawełna is feminine singular nominative, the adjective also has to be feminine singular nominative:

masculine: czysty
feminine: czysta
neuter: czyste

So bawełna jest czysta is the correct agreement.

What does jest do here?

Jest means is. It is the 3rd person singular form of the verb być (to be).

So the sentence structure is:

Bawełna = subject
jest = is
czysta = clean / pure

This is a normal Polish way to build a sentence with to be in the present tense.