There are different groups of people that respond to different sides of that message, and the people who broadcast either or both sides have differing motives as well. Countering it effectively requires understanding the problem.

The anti-welfare shtick comes from the political donor class - the capitalists, ie billionaires. Lack of a safety net gives them more leverage to negotiate lower wages within the businesses they own, and less government spending results in lower interest rates on the loans they use to grow their industries. The narrative is often framed as concerns over sharing limited resources, but the true motive is the elimination of the safety net - if enough of the working class can be convinced that the system isn't fair, that creates casus belli to delete the system.

The other side of the argument is about supply and demand of labor as a commodity. As supply goes up, the price goes down, resulting in lower wages. Its a fundamental weakness of unregulated markets operating in the context of uneven global development. Immigrants from the poorer parts of the world can accept below market rate salaries and still come out ahead, but they drive down wages of the local economy in the process. A "free" market gives the capitalists, ie billionaires, power akin to the divine right of kings - with enough capital, they can game the market to suppress competition, establish monopolies, and perpetually remain in positions of ultimate executive authority. Therefore, regulating the market to make it truly free isn't in their interest, so instead the propaganda organs they own - CNN, Fox 'News', etc... - blame the immigrants for pursuing rational self-interest as if it were a moral failing, instilling hatred in part of the working class toward other working class people to garner political support for representatives who pay lip service to anti-immigration policy. Such policy isn't actually passed - it would result in better wages and the capitalists don't want that - but the lip service wins elections in some states while also preventing the conversation from shifting to possible regulation to solve the problem of low wages.