The idea is that if you want to see whether you might like a book, the first page is not indicative. Oh, it may well turn you off a book. But, most authors who get published have learned to write a strong first sentence, first paragraph -- first chapter, even. This is intended to hook the reader. It isn't a cheat or a bait and switch, but neither is it necessarily indicative of the author's style for the rest of the book.
For that, the idea is to read a page well enough into the body of the book that the author has settled down into his or her usual style. This should give you an indication of whether you like that style.
Liking page 117 is not a guarantee that you will like the book, but not liking page 117 is a strong indicator that you will not.
This isn't always the case, and there are times when page 117 is blank or is the first or last page of a chapter. You may want to pick a new page in those cases.
I have never made it to the Page 117 panel, but I have used the test. I'd gotten used to having one of two reactions:
a) This isn't working for me. Maybe the book is badly written. More likely, it simply isn't to my taste. Either way, it probably isn't the book I want to invest my time in.
b) This could work. Nothing's turning me off. It holds my interest well enough. A lot of good writing involves avoiding bad writing.
But, not all.
Rosemary and Rue, by Seanan McGuire, and Night Watch, by Sergei Lukyanenko and translated into English by Andrew Bromfield, elicit a third reaction, one I can sometimes forget exists if I have been looking at a lot of books getting a) or b) reactions, buying books where I have sufficient trust to skip the test, or actually reading the books I have acquired.
c) I start reading. I mean really reading. In the last couple of weeks, this happened twice. For both Rosemary and Rue and Night Watch, I realized that I was not evaluating the text of page 117 to see if I might like to read the book. I was reading page 117, interested in what was going on, and wanting to know what happens next.
At that point, the best thing to do is to turn to page 1 and read the book.