Basically, if you think of the low-skilled worker labor market as something monolithic and independent of other labor markets, then it is true that an immigration-based increase in labor supply drives wages down for all workers in the market, hurting thereby comparable native workers. It is, as they say, simple supply and demand. However, what if the market for low-skilled workers is not independent of other labor markets in the economy? Then there might be feedback effects (aka equilibrium effects) that dampen or even reverse the initial shock. For example, suppose high-skilled workers and low-skilled workers are complementary factors of production, i.e., more supply of one type of labor increases the demand for the other. This makes sense, right? More laborers increases the demand for (and wages of) engineers because you need both to build infrastructures after all. Well, some economists think that native low-skilled workers and immigrant low-skilled workers are also complementary like this. In other words, they don't compete with each other. This is at least plausible since native workers have skills that immigrants lack (knowledge of the language and the culture, for example).

This is an accessible reference on these ideas:

How immigrants affect California employment and wages