First, I begin with the premise that if we are to have genuine knowledge of the world, there must be a certain isomorphism between the structures of our understanding and the real ontological structures of the things in the world which we know. In other words, our concepts of things have somehow to match and reflect and represent the way the actual things are in themselves. Otherwise we do not genuinely know about anything outside of our minds. But I take it that in the modern scientific age, we cannot seriously deny that we know things; it's taken for granted by the religious devotees of Modern Science. Not only that, the denial that we know anything cannot be coherently affirmed, unless the person who affirms it admits to not knowing whether the denial is true.
So the structures of our understanding, if we are to have genuine knowledge, must in someway match and represent the actual ontological structure of the things we know. Now, second, we find that there is a certain contingency in the way things are actually structured. For example, I have an understanding of my coffee cup as blue, cylindrical, and being made of porcelain. But not everything that's blue has to be cylindrical, nor does everything made of porcelain have to be blue, and so on. These qualities are separable from each other. The most obvious evidence of this separability is the fact that I can destroy my cup in such a manner that it might no longer exhibit all these qualities at once. And if my understanding of the cup is genuine knowledge, then the contingent unity of these qualities in my concept of the cup matches with a genuine contingency of their unity in the actual cup itself.
Now, this contingent unity has to be explained somehow. In the case of my own concept of the cup, it is my mind which brings the various component concepts of the cup into a single concept: my mind unites blueness, cylindricality, and being made of porcelain into a single concept of my coffee cup. It would seem reasonable, then, that the actual unity of the component qualities of the coffee cup (which is its existence) is itself ultimately brought about by a Mind, one which has the power to produce contingent beings and to bring them into existence by thinking of them.
Of course, the immediate efficient cause of my coffee cup was some particular artisan or whatever who molded the porcelain, painted it, and so on. But that is just to raise the same problem at a different level. That person, too, exists as a result of the unity of her ontological constituents; she exists thanks to the unity of the organic matter composing her body and its arrangement in such a manner as to produce a living, functioning human body. Because the artisan is herself a composite object capable of being understood, the contingent unity of her constituent ontological parts also has to be explained somehow.
Merely positing a long chain of composite objects which exist contingently will never explain the existence of those objects in the first place. Unless a book is first produced, it cannot be lended from one person to another, even if the chain were infinitely long, because the independent existence of the book is a precondition of the lending in the first place; an infinite chain of lenders does not and cannot explain the existence of the book. In the same way, an infinitely long chain of contingently existing composite objects cannot explain the existence of such objects in the first place.
The ultimate explain of contingent composite intelligible objects must not be any of these things. It must be necessarily existent, because otherwise it would need a cause of its own and could not function as an ultimate explanation. It must not be composite for the same reason: if it were composite, it would need something to explain the unity of its constituent parts. And finally, it cannot be intelligible because it isn't capable of being dissected into component parts and analyzed in that way. Rather, this ultimate explanation has to be the ultimate ground of all intelligibility.
This ultimate explanation must also be a Mind. It must be intelligent because it must be capable of teleological, goal-oriented behavior: it must be capable of producing particular effects and obviously a wide variety of them, as well. It must be capable, for example, of uniting subatomic particles in such a manner as to produce a hydrogen atom and not something else, and of uniting hydrogen and other sorts of atoms in such a manner as to produce planets capable of sustaining life, etc.
The intelligibility of the universe, then, is evidence of its origin in an intelligent source.