Because immigration increases the supply of U.S. labor, it reduces wages or makes jobs more scarce for natives.
This claim cannot possibly be correct over any span of time longer than the very short run.
After all, the supply of labor in the U.S. today is greater than the supply of labor in the U.S. in 1776 by a multiple of (my off-the-cuff estimate says) about 70. Indeed, the supply of labor in the U.S. today is more than double what it was in 1950. And yet there’s no serious shortage of jobs, and wage rates are immeasurably higher than they were in 1776, very much higher than they were in 1950, and, yes, higher even than they were in that golden year of 1973.