As far as I understand, the closer the points are to the line, the more distant they get to the rest of the plane. That's why he says that "this is an improper triangle", as the point of intersections of the hyperbolic lines are theoretically at an infinite distance from the "origin", and thus that the lines connecting those points have an infinite length.

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.

The disk model of hyberolic geometry is made to map hyperbolic 2 space (which is infinite in area) into the finite interior of the disk. In order to capture this, the normal euclidean notion of distance is distorted by a function which allows "distances" to go to infinity as a curve approaches the boundary of the disk.