Why Erin?

The answer is, that Éire "Ireland" has in the standard language the dative form Éirinn. A dative form is what we use after simple prepositions such as ag, ar, de, do, chuig, i and so on (but not after chun, which takes the genitive). Most Irish nouns have no particular dative form anymore, although datives are common in set expressions (the -ibh in ar na mallaibh 'lately' is a dative plural ending, for instance, and cois, chois, a chois 'beside' is historically de chois, where c(h)ois is a dative form of cos 'foot, leg', governed by de; it is also usual - to me, most natural - to use the dative form of feminine verbal nouns ending in -ach after ag or a < do, thus léimneach > ag léimnigh).

Thus:

Is í Éire tír dhúchais na nGael 'Ireland is the native country of the Gaels'

Labhraítear Gaeilge in Éirinn 'Irish is spoken in Ireland'

Poblacht na hÉireann 'The Republic of Ireland'

As you see, it is one of those pesky placenames that take the definite article in the genitive case, not in the other cases.

Anyway, in spoken dialects it's quite common to use Éirinn as nominative too. Thus, you could say:

Is í Éirinn tír dhúchais na nGael - 

and from this form, the English name 'Erin' has been borrowed.