Learn Finnish: Toinen kirja on uusi. - The other book is new.

Because they must match the noun they are connected to! The subject of the sentence is kirja (book), which is in the basic nominative case. Therefore, both the word attached directly to it (toinen) and the adjective describing it after the verb (uusi) must also be in this basic case.

I learned that toinen means "second". Why is it translated as "other" here?

Toinen is a very flexible word in Finnish that can mean either "second" or "other" (or "another"). Both "The other book is new" and "The second book is new" are correct translations for this sentence. You usually tell which one is meant from the wider context of the conversation.

How do we know it translates to "the" other book and not just "another" book?

Finnish doesn't have articles like "the" or "a/an". Because toinen means both "other" and "another", this exact same sentence could also mean "Another book is new." You just pick the English article that makes the most sense for the situation.