The collision term
$$
{\cal I}_{coll} \simeq \frac{f-f_0}{\tau}
$$
known as the relaxation time (or BGK, Bhatnagar-Gross-Krook)
approximation, is a simple phenomenological model that can
be motivated, but not derived, from more microscsopic theories
(of course, if you compute only one number, there is always a
value of $\tau$ that will give the right answer, and this is sometimes used to provide explicit formulas for $\tau$).
In general, the problem of deriving the Boltzmann equation
and the collision term from microscopic theories is very
complex. The simplest situation, dealt with in standard
text books, arises when the system is dominated by elastic
2-body scattering. Then
$$
\left.\frac{\partial f}{\partial t}\right|_{coll}
= \int dv_2 \int d\Omega\,
\frac{d\sigma}{d\Omega} \, |v_1-v_2|
\left( f(v_1')f(v_2') - f(v_1)f(v_2)\right)
$$
where $d\sigma/d\Omega$ is the cross section for $(v_1,v_2)
\to (v_1',v_2')$. This cross section can be either classical (derived from classical trajectories in a potential) or quantum (the square of an amplitude, calculated, for example, using Feynman diagrams). In the classical case this follows from the
Liouville equation and the BBGKY hierachy. In the quantum
case this was first studied by Baym and Kadanoff, and is
described in their text book. This result
is for classical particles (Boltzmann statistics), but it is
easily generalized to Bose or Fermi statistics, by adding Pauli blocking or Bose enhancement factors.
The literature contains a variety of extensions, for example for multiple
species, inelatic collisions, three-body collisions, radiation
(as in QED), etc. Deriving these collision terms from
microscopic theory becomes increasingly difficult.
Finally: How to motivate the BGK appproximation? The collision term is a non-linear functional of the distribution function,
$$
\left. \frac{\partial f}{\partial t}\right|_{coll}={\cal I}_{coll}[f] .
$$
We can define a linearized collison kernel by expanding around the equilibrium distribution $f_0$
$$
{\cal I}_{coll}[f]={\cal I}_{coll}[f_0+\delta f]\simeq L[\delta f].
$$
One can show that $L$ is a hermitean, negative semi-definite, operator. This means that we can diagonalize $L$
$$
L = -\sum_i \frac{|\chi_i\rangle\langle \chi_i|}{\tau_i}
$$
where the eigenvalues $\tau_i$ are a set of relaxation times, and the $\chi_i(v)$ are the corresponding eigenfunctions of $L$ (and I haved used bra-ket notation as in QM). If I consider the Boltzmann equation in the long-time (hydrodynamic) limit, the longest relaxation time dominates and
$$
L[\delta f] \simeq -\frac{|\chi_0\rangle
\langle \chi_0| \delta f\rangle}{\tau_0}
\simeq -\frac{|\delta f\rangle}{\tau_0}
$$
which is the BGK form.