Learn Hardcore Icelandic: Barnið elskar sig sjálft.

The ending -ið is the neuter singular definite article attached to the noun. So barn = “child” and barnið = “the child.” Here it’s in the nominative case as the subject of the sentence.

Why is it sig and not hann/hana/það?

sig is the third‑person reflexive object pronoun. It always refers back to the subject of the same clause (“the child loves itself”). To refer to someone/something else, you use the ordinary third‑person pronouns:

hann = him (masc)
hana = her (fem)
það = it (neuter, some other thing, not the subject)

What does sjálft add? Can I just say “Barnið elskar sig”?

sjálft (from the adjective sjálfur) is an emphatic “self,” roughly like adding emphasis to “itself.”

Barnið elskar sig = “The child loves itself.” (perfectly correct)
Barnið elskar sig sjálft adds emphasis: “The child loves itself (itself).”

Why is it sjálft and not sjálfan or sjálfa?

Because sjálfur agrees with the referent in gender and number. Barn is a neuter noun, so you use the neuter form sjálft (accusative here, though neuter nom/acc look the same).