The unstated, but obvious gist to begin with; is that somehow they are entitled to be placed on the same field as American citizens, which in that vein would mean they think they are better than us (having not contributed one whit, but are entitled to all of America’s promises).
That is the implied meaning of “you’re no better than me”.
That's also because they heard that "you didn't build that. Somebody else made that happen."
To foreigners, that was meant to diminish American citizens' claims on America.
-PJ