Okay kiddo, so Irish republican legitimatism is a big phrase that means some people in Ireland believe the country should be an independent republic, meaning it doesn't have a king or queen like some other countries.
These people think that Ireland's true government should be chosen by the people who live there, instead of being controlled by another country.
They believe that the British government shouldn't be in charge of any part of Ireland, and that the Irish people should have the right to govern themselves without outside interference.
Basically, they think that the only way for Ireland to be truly independent is to have a government that is chosen by the Irish people and not controlled by anyone else.