Learn Icelandic: Mig langar að kaupa eplin.

The verb langa (to want) is an "impersonal verb." This means the person wanting something isn't in the usual subject case (nominative), but rather in the accusative case. So, instead of ég (I), we must use the accusative form mig. It translates somewhat literally to "it longs me."

Why is the word for "apples" in the accusative case here?

The apples are the direct object of the verb kaupa (to buy). In Icelandic, the thing that is being bought or directly acted upon usually takes the accusative case.

How do we form the definite plural eplin (the apples) from the basic word epli (apple)?

Epli is a neuter noun. For many neuter nouns, the accusative plural looks exactly the same as the singular: epli (apples). To make it definite ("the apples"), we attach the neuter plural definite ending -n, giving us eplin.

What is the function of að before kaupa?

The word að acts as the infinitive marker, just like "to" in the English phrase "to buy." When you follow a main verb like langar with another action, that second verb stays in its basic infinitive form, preceded by að.