Learn Icelandic: Ég er frá Íslandi.

This is because of the preposition frá. In Icelandic, frá is a static preposition that forces the noun following it into the dative case. For the neuter noun Ísland, the dative form is Íslandi.

Does the dative case always add an -i to the end of a word?

Not always, but it is a very common dative ending for singular masculine and neuter nouns. Because Ísland is a neuter noun, it simply gets an -i added to the base form.

Does frá always take the dative case?

Yes! Whenever you use frá, the noun that directly follows it must be in the dative case. It is one of the most reliable prepositions in Icelandic because it never triggers any other grammatical case.

Why is it just Íslandi and not attached to a word for 'the'?

Just like in English, we say 'from Iceland', not 'from the Iceland'. The word Íslandi is the bare form of the noun, meaning no definite article is attached to the end of it.