Referring to Plasma Astrophysics, Part 1, Eq 4.32,
The number density of electrons in the beam at a distance $z$ from the injection plane at $z = 0$ is defined as,
$$ n_b(z) = \int_{0}^{\infty} \int_{0}^{\pi} f(z,v, \theta) v^2 dv 2 \pi sin \theta d \theta $$
where $f(z,v, \theta)$ is the distribution function defined in terms of
$z$ - coordinate along the direction of the magnetic field
$v$ - velocity of electrons
$\theta$ - angle between $z$ and $v$
How did this definition come about?
You are just showing the zeroth velocity moment (e.g., see answer at https://physics.stackexchange.com/a/218643/59023 for examples) of a velocity distribution function. The differential comes about because someone assumed azimuthal symmetry, i.e., you have: $$ d^{3}v \rightarrow v^{2} \ dv \ 2 \pi \ \sin{\theta} \ d\theta $$
The units of $f\left( z, v, \theta \right)$ are usually in number per unit volume per velocity cubed. So if you integrate over $d^{3}v$ you get units of number per unit volume, also known as number density.