You want to derive the Fokker-Planck equation (the drift-diffusion equation for the density) from the Langevin equation (the stochastic differential equation for the position of a particle); this is straightforward if the noise term does not depend on position, see for example these lecture notes.
The "intuition" is that the PDE must conserve the particle density, so it must be of the form of a continuity equation $\partial\rho/\partial t=-\nabla\cdot J(\rho)$. The current density $J$ will for small $\rho$ be linear in $\rho$, and if the density varies slowly it will only contain first derivatives, so $J=J_0 \rho+D\nabla \rho$$J=J_0 \rho-D\nabla \rho$. The drift term $J_0$ and the diffusion constant $D$ are obtained from the Langevin equation, by calculating the first and second moments of the displacement.
If the noise is position dependent, so the diffusion coefficient $D(x)$ is not constant, an ambiguity appears, do you write $J=J_0\rho+D\nabla \rho$$J=J_0\rho-D\nabla \rho$ or do you write $J=J_0\rho+\nabla( D\rho)$$J=J_0\rho-\nabla( D\rho)$ ? This is the Ito-Stratonovich ambiguity, which can only be resolved by a more detailed specification of the random noise, see these notes.