Quote 1
"It is simplicity itself," said [Holmes]; "my eyes tell me that on the inside of your left shoe, just where the firelight strikes it, the leather is scored by six almost parallel cuts. Obviously they have been caused by someone who has very carelessly scraped round the edges of the sole in order to remove crusted mud from it. Hence, you see, my double deduction that you had been out in vile weather, and that you had a particularly malignant boot-slitting specimen of the London slavey." […]
I could not help laughing at the ease with which he explained his process of deduction. "When I hear you give your reasons," I remarked, "the thing always appears to me to be so ridiculously simple that I could easily do it myself, though at each successive instance of your reasoning I am baffled until you explain your process. And yet I believe that my eyes are as good as yours" (Bohemia.1.12-13).
Watson's response to Holmes's reasoning is like that of a man who's just had a magician's trick explained to him: it all seems so clear after an explanation… though otherwise you'd never be able to guess how it's done. Conan Doyle repeats this formula many times to remind the reader that Holmes is always many steps ahead of everyone around him.
Quote 2
"Then, how do you know?"
"I see it, I deduce it. How do I know that you have been getting yourself very wet lately, and that you have a most clumsy and careless servant girl?"
"My dear Holmes," said I, "this is too much. You would certainly have been burned had you lived a few centuries ago. It is true that I had a country walk on Thursday and came home in a dreadful mess, but as I have changed my clothes I can't imagine how you deduce it" (Scandal.1.8-10).
"Burned"? What? Watson appears to have just called his best buddy Holmes a witch. But his point is really that Holmes's thought processes are so beyond an ordinary person's that his deductions seem like witchcraft if you don't know his methods. It's like that Arthur C. Clarke quote about all sufficiently advanced technology appearing like magic? For Watson, Holmes's intellect is like a super-advanced machine, a subject of awe and admiration.
Quote 3
My friend was an enthusiastic musician, being himself not only a very capable performer but a composer of no ordinary merit. All the afternoon he sat in the stalls wrapped in the most perfect happiness, gently waving his long, thin fingers in time to the music, while his gently smiling face and his languid, dreamy eyes were as unlike those of Holmes the sleuth-hound, Holmes the relentless, keen-witted, ready-handed criminal agent, as it was possible to conceive. In his singular character the dual nature alternately asserted itself, and his extreme exactness and astuteness represented, as I have often thought, the reaction against the poetic and contemplative mood which occasionally predominated in him (League.156).
Watson's close observation of Holmes's shifting moods proves just how much he admires the guy. But it's also a way of raising our admiration of Holmes through Watson. This passage about Holmes's love of music makes the detective seem like a complex person instead of just a brain on legs – he has thoughts and feelings that we can recognize.