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The stalk of the structure sheaf on the étale site at a geometric point is the strict henselization of the corresponding local ring, cf. Section 4 of Milne's lecture notes on étale cohomology, or EGA IV. Regarding the commutation issue raised in anon's comment: in the case at hand (Prop 6.12 of the lecture notes on motivic cohomology), $F=\mathbb{Z}_{\operatorname{tr}}(T)$ for $T$ a scheme of finite type. Because of the finite type assumption, every correspondence $\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}}\to T$ is in fact represented by a correspondence $U\to T$ where $U$ is an étale neighbourhood of $x$ in $X$; therefore the stalk of $\mathbb{Z}_{\operatorname{tr}}(T)$ at $x$ can be computed as $\mathbb{Z}_{\operatorname{tr}}(T)(\operatorname{Spec}\mathcal{O}_{X,x}^{\operatorname{sh}})$.