Learn Icelandic: Við samþykkjum boðið frá þeim manni. - We accept the invitation from that man.

The word þeim is the dative masculine singular form of the demonstrative pronoun sá (that). It is in the dative case because it follows the preposition frá (from), which always takes the dative. It is masculine and singular to match the noun it describes, manni (man).

But doesn't þeim usually mean "them"?

Yes, this is a very common trap! The word þeim is the dative plural for "them" (from the pronoun þeir). However, it is also the dative masculine singular for the demonstrative pronoun sá (that). Because the noun manni is singular, we know þeim must mean "that" in this sentence.

Why is boðið in the accusative case here?

The verb samþykkja (to accept) requires its direct object to be in the accusative case. The word boð (invitation) is a neuter noun, and for all neuter nouns, the nominative and accusative forms are exactly the same.

Why does the word for man look like manni here?

It changed to manni because of the preposition frá, which forces the following noun into the dative case. The word maður (man) is irregular, and its dative singular form is manni.