The Apple Appearance extension is a system extension for the Mac OS. It provides a generic way to define the appearance (sic!) of user interface elements like the window borders and titlebar, menus and controls, the latter being the part Coaster is using. It is the basis for graphical "themes" as introduced by Mac OS 8.5.
Do I need it?
If you have Mac OS 8 or later, you don't need to get it for yourself as it is part of the system.
If you have System 7.1 or later (i.e. 7.1.x, 7.5.x or 7.6.x), you need to download it from Apple (see below).
which you can download for free from Apple. This is the whole SDK and quite biiiiig...
Why don't you simply include it in the distribution of Coaster?
To include the appearance extension with one's product, one has to license it from Apple for an annual fee. Since Coaster is Freeware, I simply cannot afford this - sorry.