Question

Why is a world line's proper time the age of an observer moving along this world line?

Answer 1

How about looking at the evolution, with just the ordinary laws of special relativity, of, say, an harmonic oscillator attached to an accelerated box? A "clock" is not a magical device after all, its evolution can be computed (a mechanical clock would be easier to model than the full body of the observer to compute its "age", but in principle the latter could also be done

Answer 2

Explanation:

The world line (or worldline) of an object is the path that object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics.

The concept of a "world line" is distinguished from concepts such as an "orbit" or a "trajectory" (e.g., a planet's orbit in space or the trajectory of a car on a road) by the time dimension, and typically encompasses a large area of spacetime wherein perceptually straight paths are recalculated to show their (relatively) more absolute position statesto reveal the nature of special relativity or gravitational interactions.