See e.g. Section 2.1 "Talagrand's transport inequalities and Gaussian dimension-free concentration" of Gozlan's survey.
Theorem 2.3 there is Talagrand's result that the standard Gaussian measure on $\mathbb R^d$ satisfies Talagrand’s transport inequality $\mathbf T_2(2)$. On the other hand, Theorem 2.4 in that survey, with a very short proof, states that, for any real $C>0$, the $\mathbf T_2(C)$ property of a probability measure implies a concentration property for that measure.
A corollary of this result is