Unicode is essentially a giant chart of all characters. Every character in just about every language is, or at least is intended to, be represented in this giant character set. The Unicode Code Point is just the unique index number of that character.
Unicode itself is intended simply as a reference set. The actual encoding of the characters for computer can be done in many different ways. The most popular one for *nix-based systems is UTF-8, but there are others as well. HTML also has encodings for a good number of characters, for example, which is why there's an export html entry in kcharselect. Whether a character will display also depends on whether the font you use supports it, so you should make sure you install a selection of unicode fonts as well.
Now I haven't played much with kcharselect myself, but I called it up, and it looks to me like it's intended to function as a quick editor. You import a selection from the clipboard, add or edit the characters you want, and then export it back to the clipboard. There doesn't seem to be any way to directly past individual characters into another application, and the unicode number is probably just there for reference, or for jumping to a specific point in the set if you know which characters you want.
In Mozilla and Firefox, and other GTK2/Gnome applications, you can hold down the crtl+shift keys while typing the unicode code point (hex), and the respective character will appear when you release the buttons. Unfortunately this doesn't seem to be supported by KDE/QT applications yet. There may be some kind of similar function built into other apps, but I'm not familiar with them.
Also, you should be able to set up keyboard combinations that will allow you to enter various characters without needing a program like kcharselect. See the "inputting from the keyboard" section on this page for details: