The Mandelbrot set is a set but the images are not generated by evaluating some function like a sine function that traces out the border of that set and then coloring the exterior with nice colors. Instead the colors represent the escape times of points and the trajectories of those obtained by iterated evaluation of the defining function are not actually visible. Buddhabrot [1] visualizes the trajectories to some extend but more for artistic beauty than clear visualization.
Secondly there is no third dimension that you access by zooming such that each image is just a slice through some three dimensional object. Yes, one can do this and look at the Mandelbrot set as a slice through a higher dimensional object but zooming into the Mandelbrot set in your standard fractal drawing app really just magnifies parts of it. Any three dimensional appearance is just an [intended] artifact of the color palette used.
EDIT: Maybe the app used treats the escape time as height coordinate turning the Mandelbrot set into a three dimensional landscape and that is what he is trying to say. While a common idea for visualizations it is not an intrinsic aspect of the Mandelbrot set.
EDIT: I just realized that the author also wrote the fractal visualizer used to make those images. Maybe I am just misunderstanding his words or he is not great at explaining what he means or he wrote the app without a good understanding what it actually does...I don't know. Especially given that he has seemingly worked in computer graphics for a long time according to Wikipedia [2].