The first thing to understand about the principle of subsidiarity, or the principle of subsidiary function, is that it gives exactly what the label says. It is, in Benedict XVI's words, "a form of assistance to the human person via the autonomy of intermediate bodies" (Caritas in Veritate). The word 'assistance' is key here; subsidiarity is literally helpfulness (subsidium is Latin for 'help, aid, assistance'), and if we didn't like the latinate name we could simply, and with complete accuracy, call it the Helpful Function Principle or the Helpfulness Principle. It's clearly this that's front and center with the famous passage from Quadragesimo Anno:
Just as it is gravely wrong to take from individuals what they can accomplish by their own initiative and industry and give it to the community, so also it is an injustice and at the same time a grave evil and disturbance of right order to assign to a greater and higher association what lesser and subordinate organizations can do. For every social activity ought of its very nature to furnish help to the members of the body social, and never destroy and absorb them.
In other words, no state, organization, or institution should destroy or weaken the natural and (to the extent that they support common good) customary governmental, organizational, and institutional expressions of human life that are potentially vulnerable to it. It should instead help them. The most obvious example of this is the family. The family as an organization is a natural outgrowth and expression of human nature, and has as a result a natural jurisdiction, so to speak, and the jurisdictional rights that go with it. It also accumulates, depending on the society, customary privileges and jurisdictional rights. Other jurisdictions, like the state, should leave room for the ordinary and reasonable outworking of these privileges and rights; these jurisdictions have no right to replace family functions. With regard to family life the function of the state (for instance) is to help and assist the family in its functions, because doing so is one of the key ways in which the state helps "the members of the body social" and failing to do so is one way that the state commits "grave evil" against persons under its jurisdiction. Acting according to the principle of subsidiarity furthers the common good of all persons in a society; not acting according to it is effectively an attack on common good.
The subsidiarity principle is often paired with the principle of solidarity, and there is a real connection between the two. Solidarity is the active sense of responsibility of each person for each person; it therefore requires the active and free assumption of responsibility for others. Subsidiarity, on the other hand, is the assistance of actual people through intermediary organizations; it therefore requires the active recognition of the free responsibilities of others. When applied to life in general, they are closely associated with the virtues that uphold civic friendship and civic order, respectively. When applied to Christian life specifically, solidarity is a principle the purest expression of which is the Passion of Christ, while subsidiarity is a principle the purest expression of which is divine Providence, and we are called to exhibit both, in higher and purer forms than mere natural friendship and mere natural prudence require, because as Christians we are called to participate in both Christ's Passion and God's Providence.
In this light one can see that Clark is quite right to reject the interpretation in which subsidiarity is just a way of saying that smaller is better; if one took the phrase "smaller is better" rather loosely, it could very well be applied to subsidiarity, but it's also potentially very misleading. What Clarke misses, though, is that her own preferred way of speaking, "decisions should be made at the lowest level possible and the highest level necessary," runs into the same kinds of problems. Subsidiarity is no more (and no less) about height than it is about size. If we don't make too much of it (and understand "lowest level possible" and "highest level necessary" so that they end up being the same level, rather than two mutually exclusive levels), it can be an entirely reasonable approximation. But it's not what subsidiarity is about. What subsidiarity is about is recognizing what organizations express and further human personality and a truly human life in the most natural and basic and person-focused ways, and both not interfering with them to the exent that they do this and also actively furthering it. Subsidiarity will tend under common conditions to favor smaller organizations and certain levels of governance, but just as subsidiarity may actually require larger organizations to step in, or even to be created so that they can step in, so also subsidiarity may actually require that decisions be made at levels higher than necessary. Likewise, subsidiarity may at times require higher levels to make it possible for a lower level to make decisions that it would not otherwise be able to make. What really matters in subsidiarity is not size or level but active help for the true flourishing of each person through those institutions and organizations that make this flourishing possible. Clarke recognizes this point, saying that "Catholic social teaching’s principle of subsidiarity actually includes within it a strong sense of the responsibility of the government for creating the conditions of human flourishing," but does not, I think, see that this problematizes her own assumptions about subsidiarity as much as it does the assumptions she is criticizing.
It's because she does not see this, I think, that her application of the general idea to political life goes haywire. She says, for instance, "It is a mistake to approach the principle of subsidiarity within the context of the perennial American debate concerning the size and scope of government." What I want to insist is that this is simply incorrect. The American debate concerning the size and scope of government is perennial because it is one of the key debates out of which the actual governance of American society arises. Indeed, it is arguably the key debate out of which American governance arises. No principle can be applied to American government without passing through this debate. Thus claiming that it is a mistake to approach the principle of subsidiarity within the context of the debate is equivalent to saying that American society should not be governed according to the principle of subsidiarity, which both Clarke and I agree to be wrong. What we actually need in America is people from all sides of each political debate actively to communicate what they think subsidiarity implies in any given case in terms of their own political assumptions about the right scope and size of government for contributing to the good of each person in our society. In the United States we should precisely approach the principle of subsidiarity in terms of the size and scope of government (as well as in terms of every other significant debate about American governance); other societies will need to approach it in terms of the major discussions and expectations of their own societies. The reason for this is that both the customs and the overall structure of society will have a considerable influence (limited only by the limits of human nature and its needs) on what any given organization can effectively contribute to the good of the persons who compose that society, and equally considerable influence on how actual people further their good and the good of others through the institutions and organizations around them. But the principle of subsidiarity is about assisting this latter kind of activity, i.e., human persons pursuing their own flourishing and the flourishing of others through the institutions around them, and therefore how the principle of subsidiarity is applied depends very much (although not completely) on what the actual organizations of society are, and how they actually are able to operate.
Thus Clark is right that it is not about size of government. Consistent subsidiarists will support larger governmental structures where those structures will make it easier for actual people to do good for themselves and others through institutions like the family, churches, voluntary associations, etc. Consistent subsidiarists will also reject larger governmental structures where those structures make it harder for the people themselves to seek their own good and the good of others. Because there will be judgment calls on both sides, it is entirely possible to be a consistent conservative subsidiarist and equally possible to be a consistent liberal subsidiarist. For that matter, one can be a consistent libertarian subsidiarist or a consistent green subsidiarist. These political differences are different views about how the means available to a government relate to the common good. Subsidarity places only one constraint on these views: that the means actually chosen are chosen so as to help actual human persons by helping organizations intermediate between those people and their government to perform their functions in ways that help each person to flourish more. That is, it requires that the end (common good) be recognized to include the actual ability of persons to pursue excellence of life for themselves and others, and that the jurisdictional functions of intermediate organizations be respected and actively supported as a means to this end. Anything more precise will have to be worked out for each society on its own terms through the actual politics of society, within the constraints created by natural rights and the reasonable functioning of institutions like the family, neighborhoods or villages, or churches/synagogues/temples that are established by nature or by second nature as basic to that society.