I know that it is partly because the distances get smaller logarithmically as you get towards the ‘edge’, and I know how to construct a hyperbolic circle on the Poincaré disk. I just don’t have the knowledge to explain why the center is offset, and maybe a few other trends that are present in the hyperbolic circles.
Thanks so much.
Consider a diameter of the circle that, when extended, passes through the centre of the Disk. It intersects the circle in 2 points, one close to the Disk's centre, and one far (unless the circle's centre is the Disk's centre, in which case both points will be equidistant). Call these points $A$ and $B$, respectively, and the circle's centre $C$. In the hyperbolic metric, the distances $AC$ and $BC$ are the same, both equal to the circle's (hyperbolic) radius. But the segment $\overline{AC}$ is (in the Euclidean metric) closer to the Disk's centre than $\overline{BC}$, so it should appear bigger, and thus $C$ appears farther from $A$ than from $B$.
Is this enough of an explanation?