Ray starts out How to Know God Exists with some self-deprecating humor, describing himself (with examples) as a "klutz" and inviting us to wonder why we should heed the opinion, on a question of such high import as the existence of God, of someone who apparently can't perform simple household chores without risking death or maiming. My own problem, of course, is wondering why we should trust someone who assumes that "chance" is the only possible alternative to "design" as an explanation for biological complexity and adaption. Ray's answer to either question is that the answer really, really matters to us ... which doesn't, of course, have any terribly obvious bearing on the question of whether his particular answer is true.
Note that "design" is not just Ray's answer to why we have eyes and ears and livers and (presumably) why we plantaris tendons and GULO pseudogenes and erector pili muscles. It's his explanation for why we have refraction (and hence rainbows), gravity (and hence oceans and air -- since he's arguing that the planet itself was intelligently created, he doesn't consider gravity an explanation for why we have, e.g. planets and stars in the first place.
Ray repeats some familiar arguments in this chapter, with some familiar problems. He argues that we wouldn't expect a Coke can to form spontaneously, metal sheeting and labeling assembling spontaneously from simple molecules, and therefore shouldn't expect a banana or the person who eats one to originate that way. He doesn't really consider the implications of the fact that Coke cans are manufactured and cannot reproduce themselves (so cannot evolve by mutation and natural selection), whereas bananas and humans had ancestors and do experience evolution. He complains (or at least notes) that this argument was mocked, but doesn't seem to quite grasp why it was mocked (one hint: bananas themselves, as we know them, are results of human selective breeding).
In other cases, he does incorporate responses to arguments he's presented before. He notes, when making the "a building implies a builder, hence creation implies a Creator," that we have indeed seen architects and building contractors and carpenters and plumbers, and haven't actually seen a Being capable of making buildings (or bananas) out of nothing by sheer intellect, with no physical mechanism. But he moves blithely and confidently on: even a stone-age tribesman, he argues, would see that skyscrapers were manufactured and designed things. This might well be the case, though it would still imply an analogy between making mud huts and making skyscrapers; this would seem to me to strengthen the case for ascribing biological complexity and diversity to observed processes like reproduction, inheritance, mutation, selection, and drift.
Ray argues that we have no reason to ascribe a cow to evolutionary processes if we can't, ourselves, make a cow (out of nothing, furthermore). But his hypothetical stone-age tribesman could not make a skyscraper, out of nothing or even out of dirt and vegetation. That does not mean that the skyscrapers were not made by beings very like the builders and designers that the tribesman had known. By the same token, our inability to explain or duplicate every detail of naturalistic origins does not imply that we are wrong to seek explanations in terms of observed, natural causes, from gravity to natural selection.
Ray offers one further argument in this introductory chapter (of which, to be sure, fully developed arguments should not be expected; that's what the rest of the book is for): atheists are a minority. In much of the world, even people who accept evolution and a natural origin for stars and worlds are a minority. How likely is it that non-creationists are right when so much of the world is wrong? Of course, this argument has its problems: four hundred years ago, heliocentrists were a minority, and ten or fifteen centuries before that, people who thought the Earth was ball-shaped rather than flat were a minority. Evidence, not mere numbers giving uninformed assent, is relevant here.
Ray varies the appeal to the wisdom of the masses with an appeal to the wisdom of geniuses: Einstein, he assures us, believed in God. Not necessarily a personal God, not a God Who inspired an inerrant Holy Bible, and especially not a God Who judged and forgave us, but Something that Einstein thought was not quite the same as the universe itself (Einstein did not want to call himself a pantheist). Oddly, Ray doesn't present us with Einstein's arguments for God (or perhaps this is not so odd, as Einstein didn't actually present such arguments), but appeals to the authority of cosmologists as he appeals to the authority of popular opinion.