"The man has a vegetable."
Translation:Tá glasra ag an bhfear.
The extra letters are due to a thing called eclipsis. Amongst other situations, eclipsis occurs after certain prepositions, eg "ag". Also, it doesn't happen to all consonants.
It's too much information to explain in a short post. There are good explanations on initial mutations through eclipsis and lenition both here https://forum.duolingo.com/comment/10981166 and on https://en.m.wikipedia.org/wiki/Irish_initial_mutations.
That keeps muddling me to. I think the difference is that "ag" alone is a simply a preposition - and those aren't declined. "Aige" though, is a 3rd person masculine pronoun. To be exact, it's a prepositional pronoun. I think we don't have those in English and that's why we're confused.
aige isn't a pronoun, it's called a prepositional pronoun, but it is just a combined form of the preposition ag and the pronoun é.
agam, agat, aige, ag Pól, ag an bhfear. Instead of ag mé, we have agam, instead of ag tú we have agat, etc.
Tá glasra agam - "I have a vegetable"
Tá glasra agat - "You have a vegetable"
Tá glasra aige - "He has a vegetable"
Tá glasra ag Pól - "Paul has a vegetable"
Tá glasra ag an bhfear - "The man has a vegetable"
Tá glasra ag mo chara - "My friend has a vegetable"