Jury is out on this one. Japan has hardly any immigration yet their wages stagnated. Canada has a ton of immigration and their wages are also stagnant. Europe is somewhere between Japan and Canada, with also stagnant wages. Nobody disputes wages are stagnant, but the degree to which skilled immigration suppresses wages is an area where economists are unsure or even tilting towards a 'no' answer.
> if they're so good for the economy in America why aren't they good for the economy in their own country
There are so many jobs that are available only in industrialized countries. You won't find semiconductor fabs or airplane factories in undeveloped countries.