The Wikipedia article's author points out that the equations are “formally equivalent”. I think this means that both, <PERSON> and <PERSON>-Planck, equations describe the evolution of a function over time. Then, as it is done in quantum mechanics with <PERSON> path integrals, we may write the partial differential equation in terms of a path integral and talk about “propagating” the initial state through time.
Here is how to step from (0) to (1).
First, observe this property of the delta-function: $$ \int_{-\infty}^{\infty}f\left(x\right)\delta^{\left(n\right)}\left(x'-x\right)dx=\left(-1\right)^{n}f^{\left(n\right)}\left(x'\right) $$
where the superscript $\left(n\right)$ indicates the nth derivative. 916
You can show this by integration by parts.
On the right hand side of equation (0), there are first and second $x$ derivatives of $f(x,t)$ and the derivatives of $\mu\left(x,t\right)$ and $D\left(x,t\right)$ can be lumped into the $D_{1}\left(x,t\right)$ and $D_{2}\left(x,t\right)$ of equation (1). $$ \frac{\partial}{\partial t}f(x,t)=\left(-D_{1}\left(x,t\right)\frac{\partial}{\partial x}+D_{2}\left(x,t\right)\frac{\partial^{2}}{\partial x^{2}}\right)f\left(x,t\right) $$
I am not sure what happens to the term $\left(\frac{\partial^{2}D\left(x,t\right)}{\partial x^{2}}-\frac{\partial\mu\left(x,t\right)}{\partial x}\right)f\left(x,t\right)$ . Maybe these space derivatives can be ignored, if we assume that the functions $\mu\left(x,t\right)$ and $D\left(x,t\right)$ do not vary significatlly with $x$.
Now, the delta-function derivatives can be used to write the terms on right hand side as:
$$ -D_{1}\left(x',t\right)\frac{\partial}{\partial x'}f\left(x',t\right)=-\int_{-\infty}^{\infty}D_{1}\left(x,t\right)\frac{\partial f\left(x,t\right)}{\partial x}\delta\left(x'-x\right)dx=\int_{-\infty}^{\infty}D_{1}\left(x,t\right)\frac{\partial\delta\left(x'-x\right)}{\partial x}f\left(x,t\right)dx $$
and similarly for the second term. Putting it together we get equation (1).
To get to equation (2) follow <PERSON>'s hint and <PERSON> expand around $t$. 818