Each point $(n_x, n_y, n_z)$ with positive integer coordinates is one spatial standing-wave mode of an electron trapped in a box — and holds exactly 2 electrons (spin ↑ and spin ↓). Piling N electrons into the lowest-energy modes fills the positive octant of n-space out to a radius $n_\max$, the Fermi sphere. Drag the slider to grow it, then hover or click any point to inspect that mode.
$\epsilon_\mathrm{F}$ is the energy of a mode sitting exactly on the sphere's surface (Schroeder eq. 7.37). $N$ comes from the volume of the positive-octant sphere ($\tfrac18 \cdot \tfrac43\pi n_\max^3$ modes) times 2 electrons per mode (eq. 7.38).
For a real metal, $n_\max$ is enormous (order $10^7$), so this sim can't literally draw one dot per mode — it renders every mode out to $n_\max = 14$ and beyond, and the "N (continuum)" number uses the smooth sphere-volume formula instead. The "N (counted here)" row shows the same idea applied to the actual dots on screen, so you can watch it converge toward the continuum formula as $n_\max$ grows and the octant fills in more finely.
Only $n_x, n_y, n_z \ge 1$ are allowed — these are standing-wave quantum numbers, like the harmonics of a guitar string, and $n=0$ isn't a wave at all.