Learn Icelandic: Hún leyfir mér að fara.

The verb leyfa (to allow) always dictates that the person receiving the permission takes the dative case. Because of this rule, the pronoun ég (I) changes to its dative form mér.

Could I translate "She lets me go" using the verb láta instead?

Yes, but the grammar would change entirely! While leyfa means "to give permission" and takes the dative case plus að, the verb láta takes the accusative case and no að. If you used láta, the sentence would be Hún lætur mig fara.

Why do we need the word að before fara?

Here, að works exactly like the word "to" in the English phrase "allows me to go." It is the infinitive marker that connects the main verb leyfir to the dictionary form of the action verb fara.

If the sentence were "She allows him to go," how would it change?

Because leyfa requires the dative case, you would need the dative form of hann (he), which is honum. The sentence would become Hún leyfir honum að fara.