I assume you are referring to his 1914 Address to Congress. His chief evidence is that America is composed of people of diverse origins and ancestry. So, while some citizens would want one side to win, others would want another. That would create a fatal division and discord among American citizens, whose main interests should be on the side of their present homeland, that is the U.S. Basically, Wilson wants to say that he doesn't profess neutrality on behalf of his own, as a statesman and a decision maker; he professes it on behalf of American citizens and their common interest.
"Such divisions amongst us would be fatal to our peace of mind and might seriously stand in the way of the proper performance of our duty as the one great nation at peace, the one people holding itself ready to play a part of impartial mediation and speak the counsels of peace and accommodation, not as a partisan, but as a friend." - this is the strongest evidence presented in the speech. It refers to possible consequences of America's taking sides in this conflict.
