On one side, you have games such as Fate that show the players the difficulty level of a task, and perhaps upon an unsuccessful roll, the players can spend resources to raise their result. Then there's systems where, by default, the GM announces the success or failure without announcing the DC.
I know that the DC can be made public in such games, but by default and usually it's kept hidden. What is the reasoning behind this design choice? What is it trying to encourage?
It prevents metagaming. Barring magical means, there is no way to have perfect knowledge of a situation such that its difficulty can be abstracted into a number in the context of reality; those abstractions are merely a system that we (the players) use to decide what happens in the game. Some groups choose to keep DCs hidden in order to force the players to think harder about what they have to do.
This encourages in-character research rather than just applying out-of-character knowledge accrued over playing the game for a longer time. In other words, your character has to know more about the situation before you, the player, can know more about the situation and thus make the correct decision. Many groups think that the GM just telling the players a DC value makes the game easier without adding anything particularly worthwhile to the game and shifts some of the game's agency away from your character and back towards the player (which many consider inappropriate in a role-playing game).
Here's a D&D example: A rogue tries to pick a lock on a door barring the party's way and fails. The GM does not tell the party the DC of the lock they just attempted to pick, so the party considers their options. They've got a scroll of knock and a potion of cat's grace, but they don't want to waste the scroll because the wizard wants to copy it into their spellbook later, and they don't want to waste the potion just for the rogue's improved Open Lock check to not be high enough. In the end, the party decides to use the scroll because they're short on time thanks to wandering monsters.
By contrast, if the players had knowledge of the lock's DC and knew that it was only 2 greater than the rogue's previous Open Lock check, they can instantly make the optimal decision and use the potion of cat's grace to push the rogue's check high enough without wasting the scroll.
What would the characters know?
Simply put: The DC isn't something 'real' or substantial in the game-world. You can't say "Well, picking this lock has a difficulty of 15, and you did 12 lockpickyness. So, sorry champ'. A character doesn't know exactly how difficult it is. The same goes for their skills. A character doesn't actually know he has a strength of 17, for instance. However, keep in mind that characters DO know what they're good at, and that certain things are more difficult than others. For instance, they might see a lock with a DC of 50. You can tell them "This lock looks VERY difficult to pick." If they roll, say, a total of 48, you don't have to say "Nope, you failed". You can also describe it as "You can hear the lock's mechanism giving way. You're getting closer, when suddenly, at the last moment, you get stuck." Again, the same goes for skills. A character may not know that their charisma is +2 and someone else's is +5, but they MAY say that "Damn, that guy is, like, twice as charismatic as I am!"
TLDR: DC's are hidden because player-characters don't know DC's in-game. They can't technically use math to determine exactly how difficult one task is opposed to another task. They CAN, however, make well-educated assumptions. As such, you should give a more elaborate description than just 'yes' or 'no'. Instead, go with 'You got close' or 'you failed miserably' or even 'You coulda done this with your eyes closed'.
