Why thermal conductivity increases with temperature? what is the molecular mechanism with which thermal conductivity increases by increasing temperature? at least for metals? I know that heat increases the oscillations of the atoms in the crystal. But how that explains the increase in thermal conductivity? 
 A: The mechanism for increasing the thermal conductivity is phonon assisted hopping. For a disordered system, one which do not preserve the long range order, the electronic wave function becomes localized. The wave function extent is typically much smaller than the system size and is characterized by the localization length $\xi$, a parameter in theory. In this case, electron propagates by hopping events along the applied electric field. The tunneling rate for an electron to hop from a localized state $i$ to another state $j$ is proportional (exponentially) to the distance between those states $r_{ij}$ and temperature
\begin{equation}
\Gamma_{ij}=\gamma_0\exp(-\frac{r_{ij}}{\xi}-\frac{|E_i-E_j|}{T})
\end{equation} 
where $\gamma_0$ is another parameter of the theory, that depends of phonon DOS and electron-phonon coupling. From this equation you may see how $\Gamma$ (and hence conductivity) depends on $T$.
The hopping theory was first introduced to describe electron transport in disordered semiconductors. A notable person is F. N. Mott. Nowadays, it is used also, for example, for organic materials. For metal, it might be applied, if it is disordered enough, in a sense of wave function localization as said above.

I know that heat increases the oscillations of the atoms in the
  crystal. But how that explains the increase in thermal conductivity?

Yes, heat increases atomic displacements, but this apples to "normal" (not disordered) metal. As a result, electron scattering increases and conductivity decreases. Note that this is related to different conduction mechanism. If a system studied shows increasing conductivity with $T$ - it is a hopping conduction.
A classic text book on the subject is Electronic Properties of Doped Semiconductors by Shklovskii and Efros.
