Learn Icelandic: Þau þakka þeim.

The verb þakka (to thank) requires its object (the person being thanked) to be in the dative case. The dative form of "they" or "them" is þeim.

I learned that "they" can be þeir, þær, or þau. Which one is þeim connected to?

All of them! While Icelandic has three different words for "they" depending on the gender of the group (þeir for masculine, þær for feminine, þau for neuter), they all share the exact same dative form: þeim.

What does the use of þau at the beginning tell us about the group saying thank you?

Þau is the neuter plural form of "they". This means the group doing the thanking is either a mixed-gender group (for example, a man and a woman together) or a group of nouns that are grammatically neuter.

Is it a coincidence that þakka looks exactly like the dictionary form of the verb?

Not at all! For many Icelandic verbs, the plural forms (we, you all, they) look identical to the infinitive (the dictionary form). The infinitive is að þakka (to thank), and "they thank" is simply þau þakka.