"Metaphysics", for those who aren't familiar with or, worse, have a really screwed-up idea of, the term is a word that derives from the works of Aristotle. In the standard way of organizing them in the Middle Ages, there was a book called the "physics" which dealt primarily with motion. Then there was the book after the physics and, in Latin, "meta" means something like "after". So, the "metaphysics" is the book after the "physics".
However. The meaning of the name quickly got changed so that, instead of just referring to the book as a title, it instead referred to what the book was about. And Aristotle's Metaphysics discusses the fundamental nature of the world, including what it is to exist, what sorts of things exist, and how things can continue to exist and yet still change. So, that's what "metaphysics" now means: it's the branch of thought that deals with fundamental questions of existence and reality.
Now, let me be clear: there's nothing wrong with metaphysics, as long as you do it carefully. Real, rigorous philosophical metaphysics is worth studying, if only because it makes your brain hurt and leaves you questioning what actually exists, anyway. (There are reasons most academic philosophers drink.) But these are important questions and they are not answered by any science; science, after all, is only capable of talking about the world that exists, not about what makes that world capable of existing. If you wonder what atoms are like and how they work, then you're interested in doing some science; if you wonder why there are atoms at all and what it means to say that atoms exist, then you're interested in doing some philosophical metaphysics.
Religion, clearly, makes metaphysical claims. The standard Judeo-Christo-Islamic axis -- we could throw in Zoroastrianism too, of course -- claims that there is a supernatural being of unlimited power, intelligence and compassion who created us and all things that exist. If anything's a metaphysical claim, that is. It doesn't tell us anything about motion or light or any of the other standard concerns of physics; instead, it tells us how existence is possible, where existence came from, and how the world came to be at all.
This sort of approach to the metaphysical derives from a much earlier, much older tradition which saw the entire world as imbued with spiritual entities. Rivers had spirits, as did trees, the weather, natural disasters such as earthquakes, fire, and so on. The Judeo-Christo-Islamic version takes this to the ultimate degree -- instead of treating gods and spirits as a superior order of living being, God becomes the superior order of being Him/Itself. There is no longer a sort of additional species, the spiritual one, but a single being which contains all the power that was previously attributed to the pantheons of ancient gods.
Of course, this isn't the only way that religions make metaphysical claims. It's a common mistake to focus just on the Western religious traditions. These are clearly very important, having dominated history in the West for millennia, as well as significantly affecting the East. But they're not the whole story.
For example, some versions of Buddhism hold that there are no divine beings, strictly speaking. (Some versions disagree; there are strands of Chinese Buddhism, in particular, which treat spiritual beings called bodhisattvas as at least quasi-divine.) Instead, there is a process which underlies the physical world that we observe and interact with, a process of reincarnation which accounts for the existence of the living beings that we see, and for how they change and develop.
Similarly, Taoism holds, more or less, that there is a literally indescribable process -- called Tao -- which underlies all change and development in the physical world. Taoism counsels us to, effectively, surrender to this process and allow it to determine our fate.
There are also versions of Hinduism which discard literal talk of deities in favour of, again, an underlying or unifying principle -- the Brahman -- from which all existence and all things that exist emanate.
So, it would be a mistake to think that the metaphysical claims of religion are best understood as being about particular beings. They could be, as in ancient spiritual traditions, and as in the Judeo-Christo-Islamic axis, but that's not fundamental. What's fundamental is that religion, in making metaphysical claims -- claims about existence, how things come to exist, what things actually exist -- makes those claims about some other order of reality.
That is, rather than appealing to something about the natural world which makes it exist -- its necessity, its inevitability, its unity, its basic laws -- religions think that the natural world is a secondary world, which depends for its existence on another sort of world. The simplest name for this other world (and there are many names) is the supernatural.
So, that's the basic metaphysical claim of religion: that the supernatural exists, and the natural depends upon it. Unfortunately -- for the religious, although not for disreputable atheists like me -- it's nonsense.