In Vershynin's book about high dimensional probability: http://www-personal.umich.edu/~romanv/teaching/2015-16/626/HDP-book.pdf, Proposition 2.5.2 states that $X$ is a sub gaussian r.v. iff, for some $K_1$, the tail of $X$ satisfies
$\mathbb{P}\{|X|>t\}\leq2\exp(-t^2/K_1),~\forall t\geq0$
or equivalently, for some $K_2$, the moment of $X$ satisfies
$||X||_p=(\mathbb{E}|X|^p)^{1/p}\leq K_2\sqrt{p}$
The first property is evident from the pdf of gaussian, but the second property is not really clear. Though it can be shown that the moment of the standard gaussian is $O(\sqrt{p})$, it does not really tell why it is $\sqrt{p}$.