Learn Icelandic: Kakan er frá mömmu og ég tek hana úr búrinu. - The cake is from mom, and I am taking it out of the pantry.

This sentence perfectly shows the difference between the two! Use frá to talk about origin, like who sent or gave you something (from mom). Use úr when physically moving something out of an enclosed space, 3D area, or container (out of the pantry).

Why does the word for "it" translate to hana?

The pronoun refers back to kakan (the cake). Because kaka is a feminine noun, you use a feminine pronoun to say "it". Since the cake is the direct object being taken, it is in the accusative case, so the base word hún (she/it) becomes hana.

How are the prepositions affecting the words mamma and búr?

Both frá and úr force the noun that follows them into the dative case. That is why mamma changes to mömmu. For the pantry, the neuter word búr changes to búri in the dative, and when you add the definite article suffix for "the", it becomes búrinu.