It displays the visible sky at a given latitude, and relies on the Local Hour Angle of the vernal point (a.k.a. Aries, noted γ).
The first question was to re-elaborate the way the horizon is displayed..., what can be the equation of those "blue circles"?...
Use the sliders to change the latitude and Aries LHA
The applet above represents the Star Finder set for a Northern latitude (by default, you can change it). Keep in mind that you need to think you are sitting just under the zenith, figured with a blue dot in the middle of your sky.
Also, the outer blue circle is the horizon, the center one is the zenith, that means the closer to the blue center a star is, the highest in the sky it is. Remember, it is like if the user was sitting under the blue dot.
This is the opposite of what's usually used in most of the "land" sky maps, where stars are shown as seen from the ground, like is the user was sitting on the horizon, on the edge of the visible chart...
It can be confusing if you don't know it first.
The Star Finder uses the exact Local Hour Angle (LHA) to set the two discs on top of each other. The Sky Map approximates the LHA and by matching local time with the current calendar date. This obviously does not take the leap years in account, but considering the required precision, this is probably good enough.