Why a large gamma in the RBF kernel of SVM leads to a wiggly decision boundary and causes over-fitting?

The hyperparameter $\gamma$ of the Gaussian/rbf kernel controls the tradeoff between error due to bias and variance in your model. If you have a very large value of gamma, then even if your two inputs are quite “similar”, the value of the kernel function will be small - meaning that the support vector $x_n$ does not have much influence on the classification of the testing example $x_m$. This allows the SVM to capture more of the complexity and shape of the data, but if the value of gamma is too large, then the model can overfit and be prone to low bias/high variance.

which is from here(the second answer). I do understand the first part, i.e. if gamma is large, the influence of a support vector won't reach far. However, I just can't figure out why a large gamma can lead to a wiggly decision boundary and capture more of the complexity and shape of the training data,causing over-fitting thusly. Any hint will be helpful!

$$\kappa(x_i, x_j) = \langle \Phi(x_i), \Phi(x_j) \rangle$$
where $\kappa$ is the kernel function, $x_i$ and $x_j$ are data points, and $\Phi$ is the featre space mapping. The RBF kernel maps points nonlinearly into an infinite dimensional feature space.
Larger RBF kernel bandwidths (i.e. smaller $\gamma$) produce smoother decision boundaries because they produce smoother feature space mappings. Forgetting about RBF kernels for the moment, here's a cartoon showing why smoother mappings produce simpler decision boundaries: