Learn Icelandic: Barnið horfir á þá stórum augum. - The child looks at them with big eyes.

This is the instrumental dative. Icelandic frequently uses the dative case all by itself to express the means or manner by which an action is done. So, looking "with big eyes" is conveyed directly through the dative phrase stórum augum.

Auga is a weak neuter noun. Does that affect how it declines in the plural here?

Actually, in the dative plural, it behaves just like strong neuter nouns. The weak neuter body parts (auga, eyra, hjarta, lunga) have unusual endings in the singular, but in the dative plural, auga takes the standard -um ending to become augum.

Why does the sentence use the masculine pronoun þá for "them"?

The verb phrase horfa á requires the accusative case. Þá is the masculine accusative plural pronoun. This tells us the child is specifically looking at a group of males or masculine nouns. If it were a mixed-gender group, the neuter pronoun þau would be used instead.

Why does the adjective stórum end in -um?

Because it must agree with the noun augum in gender (neuter), number (plural), and case (dative). For adjectives declining strong in the dative plural, the ending is always -um, perfectly matching the noun.