Today..we just call them RINO, RATS & "Girlie-men".
Irish and German Americans were vocal in their opposition to America joining the Allies. Germans for the sake of their old country and the Irish because of England's role in Ireland. TR saw this as a danger to what he believed was a necessary American involvement in the Great War. So he attacked the concept of hyphenated Americans as those who felt America should make policy decisions based on how they would help or hurt their homeland.
Now we think of the idea of hyphenated Americans in terms of people demanding separation and special treatment for their culture. That was not the meaning 90 years ago.