Why exactly one circle?
The center O must satisfy |OA| = |OB| = |OC|. Points equidistant from A and B form the perpendicular bisector of AB; equidistance from A and C puts O on the bisector of AC.
If A, B, C are not aligned those two bisectors are not parallel, so they cross at a single point — the circumcenter. The radius is then just |OA|, so the circle is unique. On a globe the same argument runs in 3D: the center is the axis normal to the plane through A, B and C.