I am confused by the purpose of this skill [he writes]... I just fail to understand how [it] is supposed to be used in play. Here are the situations I can imagine.
There is a piece of information. It is either trivial and of no importance, or it is interesting providing some depth and background to the game and not vital, or it is a crucial piece of information.
In any possible conceivable case is the game improved by withholding any of the above information?
The answer to this question, for me at least, is a qualified "yes". There is one knowledge skill I can think of for which there is a justification, and one area in which it is concievable that the game can be improved by it. That's "Local Knowledge", or similar.
This much we know: there is a dilemma facing all DMs, especially the ones who create their own settings. On the one hand, you think your setting is interesting, and you think the players enjoy finding out about it. You also think it would really enrich the game if the players knew things that were going on, knew where things were, and knew who various NPCs were. But on the other hand, you know that your players' patience for reading or listening to infodumps about the setting are essentially nil.
This is where Knowledge skills come in: they allow you to give information to players, about the setting, that they want, at a time of their choosing. So we might imagine the following scenarios:
Bob has discovered an iron gauntlet of strange design in the dungeon. He doesn't know what it is. He rolls his "Local Knowledge" skill and the DM tells him he knows an armourer in a nearby town who could know more about it. The players go there and find out.
Gwendaline sees a strange monster she's never encountered before. After wisely fleeing, she rolls her "Local Knowledge" skill and the DM tells her of a sage who might know more. The players go to meet the sage and in exchange for performing some non-trivial task, get the information.
The party need to find an alternative route to such-and-such town because the regular road is blocked off by their enemies. Successful local knowledge rolls allow them to determine that there is a hunting trail through the forest. They go that way instead.
Hey presto! Local Knowledge makes the game better. It expands the players' knowledge of the setting in an organic way (because the information they glean is always something they actively want to know), and expands their options.
Now, of course, there's an obvious comeback that goes something like this: implicit in the above argument is the notion that for players to do or know anything, they have to roll to find out. But as -C says, it is hard to imagine how making players roll for this sort of thing really makes the game better. If the players fail their rolls, they don't get the knowledge, and the game suffers as a result, because the setting doesn't get fleshed out and the players' options don't get expanded. So why bother rolling? Why not just tell the players what they know if they ask? Or just have them ask an NPC?
There are a few reasons:
a) Having them just ask NPCs about stuff is time-consuming and quite boring. In the first scenario I included above, for instance, it would involve the players asking other NPCs they know if they know anybody who knows anything about gauntlets. Snore. Easier to roll and just say "yes" or "no", and if "no", then they can do the boring ask-other-people-if-they-know-anybody thing.
b) Having players ask the DM "What do I know about this?" or "Do I know somebody who does [x]?" or whatever puts a lot of narrative control in the DMs hands, and might prove an irresistable temptation to try to guide the "story" unless that DM is a paragon of objectivity. It might lead to scenarios in which the DM thinks to himself, "I'll tell the players this titbit of information/withold it because it'll make the story more interesting." That way madness lies. Having players roll for things like that keeps the DM honest.
c) Sometimes mystery is fun, and can make the game enjoyable in its own right. When you don't know what a monster can do, it makes it scarier. When you are carrying around some weird artefact that you don't know anything about, it makes it intriguing. When you don't know what lives in the hole in the mountain, you scout it out. Not everything ought to be mysterious, but there's nothing wrong with it in small doses.
I expect some people to disagree with this.