The cap itself lets you shrink down to the size of the epynonymous Minish. If you say "the Minish" quickly enough, it ends up sounding like "diminish", or to shrink.
The byproduct of the Minish only being seen by children: the most prominent forest-dwelling race is/are the Kokiri/Koroks, who are stated to never grow up; the largest group of Minish live in the woods. No matter how much time passed, they would always be able to communicate.
With the Triforce and Master Sword apparently having been sealed away before the Minish Cap, the Kingdom of Hyrule wasn't able to fend off a wave of monsters and demons that attacked. At least, until the Picori granted The Hero of Men Suspiciously Similar Substitutes in the form of the Light Force and Picori Blade. While they seem to be vastly weaker since they were made by a powerfully magical race as opposed to the Goddesses themselves, perhaps they used the legends of them as a basis for replacements while the real things were sealed away?
The Picori and their creations are Invisible to Adults. Fridge Logic dictates that if the Picori were truly only visible to children, they would merely be the in-universe equivalent of an urban legend. A significant number of people in the game believe that the Picori exist. Therefore the invisibility effect is either something that the Picori are doing themselves as a form of protection and can be "switched off" at will, something that can be circumvented with magic/Magitek, or both. After all, people could see Vaati just fine. Presumably the Picori's re-sizing magic (the hollow tree stumps, etc) also cancels the invisibility, meaning that the invisibility can be countered comparatively easily by anyone who can use magic.
Presumably, adults used to be able to see them when they were children themselves. That explains how everyone knows about them, though the exact way the Picori's invisibility works is still never explained precisely...
The king says that the Minish "only reveal themselves to children," so it's entirely possible that they just keep themselves hidden without using invisibility magic. And besides that, a number of NPCs, including Zelda herself, think about the Minish as a fairy tale.
At the end of Skyward Sword, Fi gets sealed in the Master Sword. What happens in the beginning of The Minish Cap? Vaati breaks the Picori Blade that looks suspiciously like the Master Sword. Since Skyward Sword takes place before The Minish Cap, the Picori Blade could very well be the Master Sword. Which means that Vaati probably murdered Fi.
Of course, the Picori Blade becomes the Four Sword, which continues to exist after other appearances of the Master Sword and has a distinct powerset; it's entirely probable that he merely killed her sister or something.
Which still means that Fi lost a sibling. Poor Fi.
Nothing suggests that the Picori Blade was ever alive, and is stated in-game to have been forged and given to the Hero of Men as a gift by the Minish.
If you run out of time before the final battle, Vaati obtains the Light Force at the cost of Zelda's life. Since he would then be omnipotent, imagine what he could do. He could have murdered thousands of people, put them all through excrutiating pain, or even destroy all forms of life withblittle effort. Especially terrifying if you think about the manga adaptation's interpretation, where Vaati stole the cap to become stronger, and was treated badly by Ezlo. Do you think he'd be above getting revenge? The answer is no.