As can be seen from the name, it is a "**local equilibrium**" distribution function. In a non-relativistic context, I mean:
$$f^0(t, \boldsymbol{x})= A(t,\boldsymbol{x})\,e^{-\frac{m}{2 K_B T(t, \boldsymbol{x})}\big(\boldsymbol{v}-\boldsymbol{u}(t, \boldsymbol{x})\big)^2}$$
where $A(t, \boldsymbol{x})$ is the normalization factor depending on local density and local temperature. 
Note that in "**global equilibrium**", $T(t,\boldsymbol{x})=T=\text{constant}$ and $\boldsymbol{u}(t, \boldsymbol{x})=\boldsymbol{u}=\text{constant}$; obviously, $A$ will Also be a constant.

The surprising point is that this function doesn't solve the Boltzmann equation you wrote: 
$$\partial_t f^0(t,\boldsymbol{x})+\boldsymbol{v}.\boldsymbol{\nabla}f^0(t,\boldsymbol{x}) \ne 0$$
However, it turns out that the left hand side depends only on $\boldsymbol{\nabla}\boldsymbol{u}$ and $\boldsymbol{\nabla}T$. 

To proceed, now we need to include the "**long wavelength limit assumption**". This is the limit of $\ell_{mfp}\ll \lambda$, where $\lambda$ is the scale over which the local quantities  $\boldsymbol{u}$ and $T$ vary appreciably. Therefore $\frac{\ell_{mfp}\boldsymbol{\nabla}T}{T}$ and $\frac{\ell_{mfp}\boldsymbol{\nabla}u}{u}$ are small quantities. This suggests to expand $f$ in a derivative expansion: $$f=f^0+f^1+\cdots$$ where $f^n\sim \mathcal{O}\big((\ell_{mfp}\boldsymbol{\nabla})^n\big)$. The local distribution
$f^{0}$ is the leading order solution in derivatives, because:
$$\partial_t f^0(t,\boldsymbol{x})+\boldsymbol{v}.\boldsymbol{\nabla}f^0(t,\boldsymbol{x}) = \mathcal{O}(\boldsymbol{\nabla})$$
To find the sub-leading solution, taking $f=f^0+f^1$, we arrive at
$$\partial_t f^0(t,\boldsymbol{x})+\boldsymbol{v}.\boldsymbol{\nabla}f^0(t,\boldsymbol{x}) = -\frac{f^1}{\tau}$$
which simply gives $f^1$. Similarly we can go through the derivative expansion to find higher order solutions $f^n$ with $n>1$.

In summary, $f^0$ describes a state close to equilibrium. The further away from equilibrium, the shorter the wavelength, and  the greater the number of terms in $f=f^0+f^1+\cdots$ to describe the fluid state.

Hope this helps.

P.S. See the kinetic theory lecture notes by David Tong: https://www.damtp.cam.ac.uk/user/tong/kintheory/kintheory.pdf.   
What I mentioned above is more or less from this reference.