Here's a great detailed primer on the issue from USA Today:
Paul Ryan visited Georgetown University, the flagship Jesuit school and decidedly hostile terrain for Ryan's strain of economic libertarianism, where he argued for his budget's priorities despite vocal and visible protests by faculty and students … sought to justify his budget priorities in terms of the Catholic principle known as "subsidiarity," but the evidence shows his platform and that principle just don't match up.
"Subsidiarity" since at least the 19th century, has been central to how the church envisions a just and equitable society functioning in a world dominated by big business and big government, both of which can dehumanize individuals and undermine the common good.
As theologian Meghan Clark noted, "is perhaps one of the most crucial and most misunderstood in Catholic social teaching." It's not just a matter of ever smaller government, or reflexively devolving responsibilities downward, but of making sure that key societal functions are provided for. "The principle of subsidiarity protects people from abuses by higher-level social authority and calls on these same authorities to help individuals and intermediate groups to fulfill their duties," says the Vatican's Compendium of the Social Doctrine of the Church.
In recent years, Catholic conservatives, and especially those like Ryan with a libertarian bent, have focused almost exclusively on the first part of the formula to present a kind of laissez-faire version of Catholic economics. In this reading, government-sponsored universal health care and social service programs, among other things, would violate Catholic teaching and infringe on the individual's freedom and duty to work hard and contribute to society.
University of Dayton theologian Vincent Miller called this interpretation the "careful lobotomization of subsidiarity," while the National Catholic Reporter's Michael Sean Winters memorably likened it to "mere subsidiarity run amok or, better to say, subsidiarity in drag."
Ryan doubled down on his version when he compared subsidiarity to "federalism," "meaning government closest to the people governs best." Even a number of Catholic conservatives were prompted to correct the House Budget Committee chairman on that score. Stephen P. White of the Ethics and Public Policy Center explained that "subsidiarity is not about exercising power at the lowest possible level so much as it is about locating social responsibility in its proper place…The goal or end of subsidiarity is the proper ordering of society for the common good."
Moreover, theologians and church leaders point out that subsidiarity without the corresponding principle of solidarity leaves people to fend for themselves, the weak falling prey to the powerful. "The right ordering of economic life cannot be left to a free competition of forces," Pope Pius XI wrote in the 1931 encyclical, "For from this source, as from a poisoned spring, have originated and spread all the errors of individualist economic teaching."
In his 1961 encyclical, Pope John XXIII articulated another element of subsidiarity, writing that, "In a system of taxation based on justice and equity, it is fundamental that the burdens be proportioned to the capacity of the people contributing."