Popper* gives us a useful definition of "metaphysical" in the Logic of Scientific Discovery. A metaphysical statement is a meaningful statement that is not in principle falsifiable by experience. Not all unfalsifiable statements are metaphysical — some are simply meaningless — but all metaphysical statements are, by definition, unfalsifiable. Note that this definition is itself metaphysical: the definition is (or at least appears to be) meaningful, but there is in principle no empirical observation we could make that could falsify it.
*I invoke Popper here not to establish authority but to give credit.
Another example of a metaphysical system is the set of rules that define the game of chess. There is no empirical observation that could, for example, falsify the rule that bishops must move diagonally in a straight line. If we observe a player move his bishop horizontally, we can conclude only that either the player has made an error or that she is not playing chess. (Note that the statement that "human beings consider chess to constitute thus-and-such rules" is a scientific statement: we can observe how human beings define chess, and in principle falsify the statement.)
Popper departs here from the Logical Positivists, the latter assert that all statements neither verifiable nor falsifiable by experience are not meaningful in any sense. Popper in contrast admits that unfalsifiable statements can be meaningful.
Popper departs as well from a common theme in philosophy, the theme of metaphysics as a synonym for ontology. In his demarcation criterion, Popper establishes a metaphysical "rule" of scientific naturalism: unfalsifiable statements are ontologically meaningless. If a statement is empirically unfalsifiable, is is for that reason categorically not a statement about the world. If it can be charitably interpreted only as looking like a statement about the world, then it is nonsense — "not even wrong" — having at best only the appearance of meaning. This principle does not deny all meaning of unfalsifiable statements, only a specific kind of meaning.
In a similar sense, the statement, "The bisectors of two angles of a triangle intersect inside the triangle," is a meaningless statement of Euclidean geometry. It's not true, it's not false. Specifically, the word "inside" is a term without referent anywhere in Euclid's axioms. We have to create a different context — e.g. analytic geometry — to make the statement meaningful and true.
Thus scientific naturalism — being itself metaphysical — is not a statement about the world. It is, in essence, a language game we play. One is free to play any language game one chooses, including religious language games and the language game of calling religious people jackasses whose views on reality and morality are at best ridiculous and at worst malevolent.
Popper's construction gives us a metaphysical framework to rigorously discuss meaningful ontological statements — i.e. statements about the world — that are not directly empirically observable. We cannot, as Hume noted, observe causality: all we can observe is that one event usually or always follows another in time. But we can falsify a causal hypothesis: We can hypothesize that event X causes event Y, i.e. that event Y will always follow event X. If we were ever to empirically observe that event Y did not follow event X, our hypothesis would be proven false; we must change something: the hypothesis itself or something in its theoretical framework.
Scientific naturalism does not deny the meaning or truth of statements that in a sense transcend empirical observation, i.e. statements whose truth or falsity we cannot directly determine by observation. Scientific naturalism not only admits statements that "transcend" empirical observation, but gives us a rigorous way of determining which transcendent statements are meaningful and a rigorous way of at least rejecting meaningful empirically transcendent statements as definitely false.
Of course, scientific naturalism does deny the meaning of statements that transcend empirical observation in a different sense, i.e. statements interpreted as about the world that cannot in principle be falsified by empirical observation.
Intelligent Design is an excellent example. At first, to their credit,
*The triviality of the proposed structures is itself suspicious.
Scientific naturalism excludes some statements as meaningless, statements that appear to have meaning, that are grammatically correct, that do indeed activate our minds in interesting and complicated ways. Perhaps it's the case that scientific naturalism is simply limited, in the same sense that Euclidean geometry is limited and cannot discuss concepts such as "inside" or "outside". There's no way to prove that scientific naturalism is not limited, that statements rejected by scientific naturalism cannot have meaning and truth in some other system.
The best we can say — and it's pretty good — is that scientific naturalism has in a couple of centuries given us a profound understanding of the physical universe from the cosmological to the subatomic, a technological civilization that can feed, clothe and house more than six billion people and has at least the potential for real humanistic justice and universal prosperity, and is beginning to crack the mysteries of consciousness and human behavior. In contrast, after more than two millennia religion has given us nothing but mystical mumbo-jumbo, ridiculous self-serving and self-aggrandizing fairy tales, repression, oppression and the near-constant support of even the most monstrous and abhorrent ruling classes that would maintain the privilege and status of the priesthood.