What the two pictures have to do with each other
Both pictures iterate z ↦ z² + c. On the left, z starts at 0 and you vary c:
the black plate is every c whose orbit never runs away — the Mandelbrot set.
On the right, c is frozen and you vary the starting z: that is the Julia set
for this c. The Julia set is connected lace exactly when c sits inside the
Mandelbrot set, and shatters into Cantor dust the moment c leaves. The red
polyline is the critical orbit of 0, the witness that decides which.