Learn Icelandic: Þau koma úr búðunum. - They are coming out of the shops.

The preposition úr (out of, from) always forces the noun that follows it to take the dative case. Because they are coming out of the shops, the word for shops must be in the dative.

How can I tell that búðunum means "the shops" and not just "shops"?

You can tell by the ending! A bare plural noun in the dative case almost always ends in -um, so "shops" would be búðum. When you want to say "the shops" (definite), the ending becomes -unum. That is why we use búðunum here.

Why use Þau for "They" instead of Þeir or Þær?

Þau is the neuter plural form of "they". In Icelandic, you use þau when referring to a mixed group of males and females, or a group of neuter nouns. If the group was made up entirely of men, you would use þeir, and for a group of only women, you would use þær.

Could I use frá instead of úr to say they are coming "from" the shops?

You could, but it changes the meaning slightly. úr specifically means "out of" an enclosed space, meaning the people were physically inside the shops and are exiting. frá is used for general movement away from a place or starting point without implying they were ever inside.