I'm an Irishman
Leather weathered Irish skin

from Macklemore and Ryan Lewis – Irish Celebration Lyrics on Rap Genius

Meaning

He opens saying he’s an Irishman because he’s proud, like many Irish American are, of their heritage. He also says ‘Irishmen'in an Irish accent to be patriotic.

Leathered skin is a play on the common phrase “thick skin”. To be thick skinned means that someone can take a lot of shit. This is true for the Irish considering they have a rough history both in Ireland and in the North Eastern United States. Macklemore is proud of his ancestors for overcoming these hardships.

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