"Wedge issue" politics, I think.
I don't know for sure why (SOME) poor people vote Republican. I'm not poor, though I'm not rich either, and the last time I even thought of voting for the GOP was in 1964, which was before I was old enough to vote.
Buy what some apparently intelligent and truthful pollsters & the pundits suggest is that some poor people are cultural conservatives -- some on the issue of race, some on the issue of sexual morality, some on whether religion should be an accepted part of American politics and be taught in the public school. Some poor people are strongly nationalistic and patriotic, too, and generally support the US military no matter what it does.
Many though not all Democrats, for what I think are good reasons, have generally chose to advocate for black people and other nonwhite minorities, for gay rights, and for a strict separation of church & state in American life. Many Democrats -- following an example set by a good number of Republicans, back in the Nixon administration -- also support a woman's right to legal abortion.
Many though not all Democrats -- following the example of a FEW Republicans in the 1960s -- also have been highly critical of what they see as unwise and unnecessary American wars - for example, in Vietnam in the 1960s and 1970s, and in Iraq in 2003.
According to such writers as liberal journalist E.J. Dionne, author of Why Americans Hate Politics, and Kim Phillips-Fein, author of "Invisible Hands: The Growth of the Conservative Movement from the New Deal to Reagan," Republicans since about 1970 have made a point of appealing to poor, culturally conservative ex-Democrats by taking a conservative line on racial integration, school prayer, abortion, women's rights and gay rights in general, and US foreign policy.
That has helped to convert some poor Americans, especially poor evangelical whites, into Republican voters, although they used to favor the Democrats.
-- democratic socialist