Learn Hardcore Russian: Учитель объясняет задачу.

Учитель is a masculine noun in the singular. In Russian, most nouns ending in a consonant (here: -ль) are masculine. You also see agreement with the verb: учитель объясняет uses the 3rd-person-singular ending -ет, matching a singular subject.

What is the case of задачу, and why is it used here?

Задачу is in the accusative singular. It denotes the direct object of the transitive verb объяснять (“to explain”). Feminine nouns ending in -а in the nominative (задача) change to -у in the accusative.

Why is there no preposition before задачу?

Transitive verbs in Russian take their direct objects in the accusative case without a preposition. English “explain something to someone” splits into two parts, but Russian simply uses accusative for что? (“what?”) and dative for кому? (“to whom?”) with no extra word like “to” before the object.

What is the verb form объясняет, and what does it tell me?

Объясняет is the 3rd-person-singular present-tense form of the imperfective verb объяснять (“to explain”). It indicates that the action is ongoing or habitual (“he/she explains”). The perfective counterpart is объяснить, whose 3rd-person-singular future form is объяснит (“he/she will explain”).