Learn Finnish: En osta kylmää kahvia. - I do not buy cold coffee.

This is adjective agreement! In Finnish, adjectives must match the case of the noun they describe. Because kahvia (coffee) is in the partitive case, the adjective describing it must also be in the partitive case. So, the base word kylmä adds an extra -ä to match.

Why do the words for "cold coffee" need to be in the partitive case here?

The main rule at play here is that in negative sentences, the object of the action is always in the partitive case. Because you are not buying the coffee (en osta), the noun kahvia takes the partitive ending, and its adjective follows suit.

Why is the verb osta instead of ostan (I buy)?

In Finnish, the negative word is actually a verb that changes depending on who is doing the action. Here, en means "I do not". When you use a negative verb, the main verb drops its personal ending. So, ostan loses its -n and becomes the base form osta.

They are in the same case, so why does kylmää end in -ä but kahvia ends in -a?

This happens because of vowel harmony. The word kylmä has the front vowels y and ä, so its ending must use the front vowel -ä. The word kahvi has the back vowel a, so its ending must use the back vowel -a.