An identity for the Laplace transform of a random variable and Isserlis theorem

Question

In this post there is a clever proof of Isserlis' theorem, but it is based on a identity for the Laplace transform that I do not know how to prove. Why is it that, given a random vector $X,$ it holds that $E(e^{s^\prime X})=e^{\frac{1}{2}s^\prime C s}$? (Here, as in the above-mentioned post, $s^\prime$ is the transpose vector of $s$ and $C$ is the covariance Matrix of $X$)