Okay, kiddo, have you ever heard someone say a certain word that isn't in English but you know what it means because they told you? That's kind of what an exonym is - it's a name for something in a different language than the one you usually speak.
Now, in Ireland, there are two main languages: Irish (which is also called Gaelic) and English. And because people have been speaking these languages for a very long time, there are lots of different words that mean the same thing.
For example, have you ever heard of the city called Dublin? That's what most people in the world call it, but in Irish, the name is actually "Baile Átha Cliath". That's a pretty different name, right?
So an Irish exonym is just a word that people in another language use to refer to something in Ireland, instead of using the Irish word. It might be because they don't speak Irish themselves, or just because they've always called things by a certain name and it's hard to change. Either way, it's important to remember that there are different names for things depending on who you're talking to!