The answer to the question of how a thinking, rational adult can be religious is simple. The majority of human beings who have ever lived have been religious, and I am not aware of any compelling evidence that either thinking or rationality are uncommon traits among human beings, so there does not seem to be too much of a trick to it. Even an atheist like me can see that. In fact, the common error committed by all of those who seek to drive the stake of rationality through the heart of God is to imagine that reason and faith are even opposing principles at all. They are not.
This may be true, however, mostly on a theoretical level. I will seek to demonstrate that in principle reason and revelation can coexist perfectly happily. But the problem that most perturbs the modern atheist is not this theoretical one, but the practical one of how they structure their polities and societies when they must include in them these religious people whom they see as a dangerous source of irrationality. This is why so many attacks on religion focus on particular practices by particular religions which reek of irrationality, even though the atheist critic often has little understanding of that which he criticizes. It is easy, and also largely irrelevant, to point out that the world is not really four thousand years old. It is much harder, and of the utmost relevance, to deliver a killer blow to the principle of faith. Here our rationality wholly fails.
To understand this, we need first to understand what science is, and what religion is. I will do this by viewing religion from the perspective of science. Science rightly claims a vast corpus of knowledge of "how things work", and it does not like religion to make claims in that well-furrowed ground. To scientists, these are the most noticeable claims made by the religious, not to mention the most outrageous; the scientists have investigated how the world works, they are happy in their knowledge, thank you very much; they do not intend to take the word of a long-dead scribe. But religion makes very few claims in this area. It is concerned with the world above, beyond, below, not the one here.
Religion may once have held a position where it was lord and master over our understanding of physical facts, too. The Catholic Church tried and spectacularly failed to suppress Galileo's science, as no atheist will ever allow us to forget. Opponents of evolution are another well-worn target of atheist anger. Yet such behaviour is in no way a logical extension of religious belief. Science flourished in Islam's classical period, and the men of the Enlightenment were men of God: to claim that they were atheists, as many now do, is an absurd anachronism. They simply had a much greater understanding of the possibility of reason and revelation coexisting than our modern atheists do.
It is noticeable that religious characterizations of the physical universe, insofar as they exist at all, target the past: science is far too competent at exposing the truth about the present to allow anything else. As it is, it is clear that religion has almost entirely ceded its claims on this area to the scientist, which makes the latter all the more zealous in pressing for his total victory. Within his realm of inquiry, good luck to him. But this perspective of the scientist does not allow us to grasp what religion truly is; after all, it existed before modern science, and is largely unconcerned with it. We must enlarge our view to grasp what it is truly concerned with, and hence what religion truly is.
By doing so we can see that reason and faith can coexist, and hence how a thinking, rational adult can be religious. Religion's primary concern is not with the brute facts of the world but with the meaning that we invest in these facts. It exists above, beyond, behind, and not in these facts, which is where science dwells. Facts are the things on which the categories of truth or falsehood operate: a fact is true because it can be proven it is not false. To deny a fact is not the act of a thinking, rational adult. But the meaning invested in a fact varies according to each and every thinking, rational adult.
Imagine a woman whose lover is shot as he storms Omaha Beach. The bullet pierces his helmet, which gravity had pulled down just over his left eye, blocking his view of the gun position for just long enough; the helmet was useless, structurally weakened by a prior impact, as forensic science could tell us, if it is ever recovered from this godforsaken beach; the bullet enters his eye, it pierces his brain, and he is gone. Gone but not forgotten. To the Nazi who killed him, it was a good shot, a good act: it strengthens his faith that he will succeed in defending what he believes in, which is for him the meaning of this act. To our man's lover, it is a tragedy, the beginning of a lifetime of sadness, the end of her world, the denial of a future full of the meaning derived from her happy family. A Christian in India reads of the death in a newspaper and wonders merely at man's continued inhumanity to man, and the futility of war.
Reason, while wholly capable of explaining the facts of what happened by its application of science, is mute on these meanings. It does not understand them, much less voice one of its own. It does not understand the passions of the Nazi, the lover, or the Christian. For reason is always a means to an end, and that end is always outside of reason. One realizes the absurdity of our title question when one considers that both thinking and rationality can be used to construct a theocracy by a man merely thinking how to apply the principles of a religious text rationally to politics. Both thinking and rationality will do what they are told, because they are in the thrall of what really matters: our capacity to appreciate meaning. Religion is one of the things that gives life meaning, whereas rationality and thinking are processes whose meaning is always outside of themselves.
Rationality can no more disprove the claim of the Christian to understand the true meaning of an event than it can disprove that I love my mother, or prove that our Nazi is wrong. We need something else over and above thinking and reason to do this, because neither are sufficient to understand either meaning or morality. (Is it even right that I love my mother? What if she is a torturer?) It is appropriate to talk of rationality when designing an efficient combustion engine, because we are dealing with physical facts that we understand. It would not be rational to make it out of chocolate. But it is inappropriate to talk about rationality when we are dealing with religion, because religion is primarily a way that people give shared meaning to their experiences of (to give just some examples) love, death, suffering, and joy.
A Christian understands why, if one is forced to make combustion engines for Nazis, it would be better to make them out of chocolate. I'm sure our atheist agrees with this sentiment, citing secular standards of justice or morality, but his faith in these standards has no more to do with rationality or thinking than do the Christian's faith in his own. A rationality unhinged from these standards simply performs the manufacture, because it does not understand the meaning of its act; the Christian and the principled atheist refuse, they are tortured, they die; they are remembered and the invocation of their names reminds us of the profound meaning of martyrdom.
To summarize: what I have argued here is that those who question religion on the basis of rationality are muddled in their understanding of what religion and rationality are. To someone with religious faith, it is perfectly rational to be religious; to someone without it, the opposite is true. This is obvious. The fact of religious faith is completely external to the operation of reason, which begins after faith has been established or denied. The modern atheist is engaged in a process of shouting, ever louder, "I do not believe!", when this is nothing but a statement of the obvious. They cannot prove by any mathematical formula, series of logical statements, or any of the physical facts of reality that God does not exist. The religious cannot prove that he does exist by any of the same methods. Faith does not involve rationality.
Western politics, especially in the United States, often seems to have been a pitched battle between the self-consciously rational atheist on one side and the self-consciously anti-rational Christian on the other. The question, "how can a thinking, rational adult be religious?", is a rhetorical trope in this battle, not one that comes from someone who has thought about the true nature of either reason or faith. The purported battle between the two is not determined by either faith or reason properly understood. Rather, it is the result of the rise of a faith in rationality and atheism as agents of positive change which see religious faith as their enemy: a new understanding of the meaning of life, which naturally target the old to discredit it. It cannot "prove" itself except insofar as it demonstrably moves men and gives meaning to their lives. It must compete in this realm. It is still a faith.