Learn Hardcore Latin: Mater statim ianuam claudit. - Mother closes the door at once.

Because mater is the nominative singular form, used for the subject of the verb. In this sentence, the mother is the one doing the action (claudit = shuts/closes), so Latin puts mater in the nominative.

How do I know that ianuam is the direct object, and why does it end in -am?

Ianuam is accusative singular, which is the case most commonly used for the direct object—the thing being acted on. The verb claudit (closes) needs an object: she closes something, and that something is the door. For a 1st-declension noun like ianua (door), the accusative singular ends in -am:

nominative: ianua (door)
accusative: ianuam (door, as object)

What tense is claudit, and how can I tell?

Claudit is present tense, 3rd person singular, active: (she) closes/shuts. You can tell by the ending -it, a common 3rd person singular present ending (especially in 3rd-conjugation verbs like claudere).
So claudit = he/she/it closes.