Obviously the Democrats get the votes. So why the support from Republicans? Because big business interests support "immigration reform." The biggest push is from tech companies, so they can bring in more H1-B visa workers. They can replace American tech workers with H1-B visa workers--who still have to be trained at their jobs. (See also here). The advantage for the companies--they can pay the foreign worker's less salaries and, because they risk losing their visas, the foreign workers can't jump ship. Just a new form of indentured servitude.
Businesses that employ low-skilled labor also benefit, because an influx of immigrant labor also pushes down labor costs.
Benefit to the middle-class? None that I see.