Let us define the hyperbolic plane to be the set $\mathbb{H^2} = \{x \in \mathbb{R^2} \mid |x| < 1 \} = \{(x, y) \in \mathbb{R^2} \mid x^2 + y^2 < 1\}$
In my textbook it states that in the hyperbolic plane, hyperbolic length and area are measured infinitesimally in terms of euclidean length and area by
$d_H L = \frac{2}{1 - |x|^2} d_E L$
$d_H A = \frac{4}{(1 - |x|^2)^2} d_E A$
Can anybody explain why this is? No explanation was provided in my textbook.
Think of the unit disk as covered with sticky stuff that's harder and harder to walk through the nearer you are to the boundary. In fact as you approach the boundary along a radius walking at what seems to you a steady pace, expending constant effort, you are in fact covering distance at a slower and slower rate. You can never get to the boundary.
If you want to walk from one interior point to another, the fastest route starts out taking you toward the center for a while, and then back. In fact it's a circle through those two points that is perpendicular to the boundary. Those are the "straight lines" in this model of hyperbolic geometry.
Here's a picture from https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model
Instead of thinking about moving more slowly near the edge, you can think of moving at a constant rate while distances are expanding. The differential for the hyperbolic length in terms of the Euclidean length quantifies just how sticky the stuff is, so how the distance expands. The slowdown factor is inversely proportional to the distance from the boundary: $2/(1-|x|^2)$.