Spin orbit coupling is a relativistic correction to the energy levels of an electron orbiting a nucleus. It tells us that there is an energy contribution that is proportional to $L\cdot S$, where $L$ is the angular momentum relative to the nucleus and $S$ is the spin of the electron. The intuitive picture for the Kane-Mele term goes like this:
Imagine the hopping of an electron from site $A$ to site $B$. We will view this in a path integral type picture, where the electron takes all possible paths from site $A$ to site $B$ and sums the corresponding amplitudes. The majority of the total amplitude comes from paths close to the saddle point path -- close to the straight line connecting $A$ and $B$.
As the electron follows a particular path, it interacts with all the nuclei in the honeycomb lattice of graphene. In particular, let us focus on the spin-orbit $L\cdot S$ terms coming from all the nuclei.
Nearest neighbor case
Now, let us assume $A$ and $B$ are nearest neighbors. We will keep in mind that the lattice has a reflection symmetry where the mirror lies on the line joining $A$ and $B$. Imagine a spin $\uparrow$ particle takes a spacetime path $\mathcal{P}$ from $A$ to $B$. Let this path have an action associated to it $S_\uparrow[\mathcal{P}]$. The total probability amplitude for hopping of a spin $\uparrow$ particle is given by the path integral
$$ \Psi_\uparrow(A\to B) = \sum_{\mathcal{P}} e^{iS_\uparrow[\mathcal{P}]}$$
Now consider the opposite spin and the reflected path $\mathcal{P}'$. Reflection symmetry tells us that the SOC contribution from an ion $C$ in the lattice to is $S_\uparrow[\mathcal{P}]$ is exactly equal to the SOC contribution of $C'$ (the reflected counterpart of $C$) to $S_\downarrow[\mathcal{P}']$ -- to see this just draw an arbitrary path from $A$ to $B$ and pick an arbitrary ion $C$ and look at $L\cdot S$ at an instance in the path; repeat for opposite spin with $\mathcal{P}'$ and $C'$. The non-SOC contribution to the action should also be same for $\mathcal{P}$ and $\mathcal{P}'$ (regardless of spin) due to reflection symmetry. Therefore, taking into account the SOC contribution from all ions $C$,
\begin{align}
\Psi_\downarrow(A\to B) &= \sum_{\mathcal{P}} e^{iS_\downarrow[\mathcal{P}]} \\
&= \sum_{\mathcal{P}'} e^{iS_\downarrow[\mathcal{P}']} \\
&= \sum_{\mathcal{P}} e^{iS_\uparrow[\mathcal{P}]} \\
&= \Psi_\uparrow(A\to B)
\end{align}
that is, the hopping amplitude for both spins must be exactly the same. The Kane-Mele term is not allowed.
Next-nearest neighbor case
If $A$ and $B$ are next-nearest neighbors, the symmetry that we take advantage of (with mirror axis on the line joining $A$ and $B$) is not present anymore. The two spins can have different hopping amplitudes. This allows for the Kane-Mele term. It is also intruitively clear that there is one ion $C$ that will have the maximal SOC contribution, and there is no equivalent ion $C'$ to cancel this contribution.
