Learn German: Der Mann kauft das Haus. - The man is buying the house.

Because the subject is der Mann, which acts like er (he). For regular verbs in the third-person singular (he, she, or it), you take the verb stem (kauf-) and add -t to get kauft.

The translation says "is buying". Why isn't there a word for "is" in the German sentence?

German doesn't have an "-ing" form (continuous tense). You just use the simple present tense. Der Mann kauft translates to both "The man buys" and "The man is buying". Saying Der Mann ist kauft would be incorrect in German.

Does the verb kauft always have to go right after the subject?

Not necessarily right after the subject, but it must be in the second position of the sentence. This is the German V2 rule for statements. Here, the subject Der Mann is the entire first idea (position one), so kauft takes position two. If we started the sentence with "Today" (Heute), the verb would still come second: Heute kauft der Mann das Haus.

Since the house is being bought, shouldn't its article change to show it's the direct object?

It is the direct object (in the accusative case), but neuter nouns like Haus use the article das whether they are the subject or the object. Only masculine nouns change their word for "the" (from der to den) when they are the object.