Yes, yes, yes. Science is a method. It is a means to understanding. It is a point of view. It is not a collection of information you can download into your cerebellum.
With that in mind, I’m going to suggest The Map that Changed the World by Simon Winchester. It’s about William Smith, a guy who in the late 18th and early 19th century became curious about the layers of rock underlying England, and who made a career out of surveying possible dig sites for canals and other similar works basically as a means of funding his own personal investigation. His end result was a spectacular map of English strata, and in so doing, he laid the groundwork (pun only partially intended) for modern geology. Winchester occasionally overstates his case somewhat, pumping up Smith’s contribution for the sake of drama, and his writing style is occasionally pretty flowery, but it’s a fascinating biography nonetheless.
The reason I’m suggesting this one, rather than a “pure” science book like the Dawkins or Feynman titles (which are also worthwhile), is that as a new visitor to the land of science, I think it’s worthwhile to remember that science is done by people. They are men and women who are curious about the world, and who employ various devices from the scientific toolkit to take that world apart in order to make sense of it. The Map that Changed the World is a biography of an interesting man who led an interesting life, and who contributed greatly to the science of geology along the way. Since you asked how it’s possible that we know some of the things that we know, it’s valuable to focus in on a critical moment and take a ringside seat as an individual scientist begins to formulate his hypothesis, then laboriously collects his facts and collates them into a coherent theory.
If you enjoy that, I’ll also suggest you track down a copy of Privileged Hands, the autobiography of paleontologist Geerat Vermeij. It’s somewhat hard to find (Amazon says it’s out of stock), but it’s definitely worth reading. The reason I’m pointing to this title in particular, and the reason it should interest you as a lay reader, is that Vermeij has been completely blind since the age of three, and despite that fact has become one of the world’s foremost authorities, and possibly the authority, on the evolution of mollusks. His story has something of the “inspirational tale” about it, which should appeal to the part of you that still longs for spiritual uplift; it’s fascinating how Vermeij transforms what should be an obstacle into a definite asset, as his sensitive fingertips trace out distinctive features on his fossil samples that other scientists, relying on a conventional visual examination, have completely missed. But over the course of the story, he becomes just another scientist, following the dictates of his curiosity and assembling a consistent, coherent view of his subject based on the scientific tools at his disposal, and his autobiography turns into something like a treatise, aimed at the lay reader, on the topic; and most interestingly, he himself would deny that his use of senses other than sight gives him an alternate advantage. Really, he’s just a scientist, doing what scientists do. (For an interview in which Vermeij discusses some of his views on evolutionary biology, and in which the subject of his blindness never comes up, see here.) Again, the point is to see that science is the result of work by people, in all their flawed glory.