Learn Russian: Автор пишет книгу.

Книгу is the direct object of the sentence—the thing being written. In Russian, direct objects take the accusative case. For feminine nouns ending in -а like книга, the ending changes to -у in the accusative case.

This lesson is about ANIMATE objects, and an author is a living person. Why doesn't автор change its ending?

Even though the author is animate (a person), автор is the subject of the sentence—the one doing the writing. The special "animate accusative" rules you are learning only apply to the direct object (the receiver of the action). Subjects always stay in the nominative (dictionary) case.

If книга (book) is INANIMATE, why did its ending change? I thought inanimate objects don't change in the accusative case.

That rule only applies to masculine and neuter nouns! For feminine nouns ending in -а or -я, the ending always changes in the accusative case (-а becomes -у), regardless of whether the noun is a living thing or an inanimate object like a book.

The dictionary form of the verb 'to write' is писать. Why does it change to пишет with a ш?

Писать is a verb that has a consonant change when it is conjugated. The с in the stem changes to a ш for all present tense forms, which is why 'he/she/it writes' is пишет. You just have to memorize this spelling change for this verb!