It's a bit analogous to the way train tracks shrink toward the horizon and make an angle with each other where they appear to meet it, even though they don't actually meet in the plane. These hyperbolic lines won't actually ever meet in the hyperbolic plane either but they approach the same point on the horizon.

That edge is basically an artifact of the model, you can equally model the hyperbolic plane space as a disk and then the boundary is a circle, or on an actual hyperboloid in 3D and it extends out forever.