Translation:She has a book and he has a newspaper.
The hint says that tá ... aige is "he has". Irish doesn't have a verb for "have/has" - it uses the tá ... ag construction instead, which literally means "... is at". Tá is the present tense for of the verb Bí (giving us "is" in English).
Tá leabhar aici - "a book is at her" - "she has a book". aici comes from ag sí.
Tá nuachtán aige - "a newspaper is at him" - "he has a newspaper". aige comes from ag sé.
There is more detail in the Tips Notes for the Phrases skill.
Verbs don't have gender in Irish. And they don't have the verb "to have". Instead, they say "to be at one". So in English we say "she has a book" but in Irish they say "a book is at her". But Irish grammar is different, so it's more "is a book at her".
aici and aige are prepositional pronouns, a fusing of the preposition with the pronoun.
There are lots of sites that provide tables of prepositional pronouns:
These are some of the examples that showed up in a quick search.
I wasn't suggesting that you don't know how to search, I was providing additional information as you said that the original sources were unavailable.
I replied to your post, rather than to Conor697532's post, to acknowledge that you had provided the primary answer, and I was providing supplemental information, rather than competing information.
You can hear leabhar pronounced in a number of different exercises on Duolingo:
Leabhar leabharlainne https://www.duolingo.com/comment/8504547
Léann tú an leabhar leis https://www.duolingo.com/comment/4288896
Léann sí as leabhar é https://www.duolingo.com/comment/23025621
Leabhar a mic https://www.duolingo.com/comment/4800542
??? Tá is a verb - it doesn't have gender. aici and aige are prepositional pronouns, not verbs.
Irish doesn't have a verb for "to have" - it uses the tá .... ag construction instead. There's a little more detail in the Phrases skill
Tá X ag Y is literally "X is at Y", but what it means is "Y has X"
When Y isn't a proper Noun, like Sean or Maire, but is a pronoun like "you"/tú or "he"/sé, you end up with "ag tú" becoming agat, "ag sé" becoming aige, and "ag sí" becoming aici.
That's strange. Although there is nothing wrong with "She has got a book", and it is synonymous with "She has a book", "She has a book" is preferred in more formal/educational/professional/business settings.
Next time something like that happens, flag it and report "My answer should have been accepted."
It's a waste of time flagging and reporting "My answer should have been accepted" in this case, because the course DOES accept "she has a book" - just look at the sentence at the top of this page.
If you are getting unexpected behaviour in Duolingo, you have to bring it to the attention of the Duolingo engineers, by submitting a bug report, including a screenshot that demonstrates the issue and a detailed listing of the platform and version numbers involved. You're probably not going to be much more successful than you are reporting it to the Course Contributors, but at least it is technically possible for the Duolingo engineers to fix this kind of issue, something that the Course Contributors can't do.