Learn Finnish: Tarvitsemme lisää puhtaita pyyhkeitä.

Two reasons! First, the word lisää (more) always requires the following noun to be in the partitive case, because it refers to an undefined or uncounted amount. Second, since you are asking for "towels" (plural) rather than just one towel, you must use the partitive plural.

How does the word puhdas (clean) turn into puhtaita?

Puhdas ends in -as, so its vowel stem is puhtaa-. To form the partitive plural, we need to add the plural marker -i- and the partitive ending -ta. When we add -ita to the stem puhtaa-, one of the a vowels drops out so we do not end up with three vowels in a row, giving us puhtaita.

How do we form the partitive plural pyyhkeitä from the basic word pyyhe (towel)?

Pyyhe is a word that ends in -e. These words have a stem ending in a double e, and they also use the strong consonant grade. This makes the stem pyyhkee-. When we add the partitive plural ending -itä, one of the e vowels is dropped, resulting in pyyhkeitä. Notice that we use -itä instead of -ita because of vowel harmony (the word contains a front vowel, y).

What is the dictionary form of the verb (we need)?