Learn German: Der Mann schreibt das Buch. - The man is writing the book.

The accusative case only changes the masculine definite article (der becomes den). The neuter article das and the feminine article die stay exactly the same whether they are the subject (nominative) or the direct object (accusative). Since Buch is a neuter noun, it stays das Buch.

How do we know Der Mann is the subject doing the writing?

The subject of a sentence is in the nominative case. For masculine nouns like Mann, the nominative article is der. If the man were the direct object of the sentence, the article would change to the accusative den Mann. Because it says Der Mann, we know he is the one doing the action.

The English translation says 'is writing', but where is the German word for 'is'?

German does not have a present continuous tense like English does (is writing, are playing). The single verb schreibt can mean either writes or is writing, depending on the situation. You never translate the 'is' into German for this tense!

Why are Mann and Buch capitalized?

In German, every single noun (a person, place, or thing) is capitalized, regardless of where it appears in the sentence. Since Mann (man) and Buch (book) are nouns, they must start with a capital letter.