Generated using the following logic.
Begin at a center starting point and draw two one-unit line segments from the center point in opposite directions.
Draw two one-unit line segments that begin at the end of each of the previous step's endpoint, perpendicular to the previous segment and opposite one another. Continue doing this while a given path does not terminate.
A path terminates if both new segments would touch another path from this or a previous step.
Here is a visual of the first few generations, center point and terminal points marked.
I believe this is equivalent to the toothpick sequence, OEIS A139250.
There are many comments, connections, and conjectures listed there.
This image shows the construction of the toothpick sequence:
Each "toothpick" is the same as two of your line segments, and is added with its midpoint at the end of a previous toothpick. If you delete the halves of toothpicks that run into previously or simultaneously placed toothpicks, it should look the same as your picture.
The OEIS also has code to generate animations such as this one: