Heuristically, this result follows because the semiclassical wavefront set measures oscillations at frequency $\frac{\xi}{h}$. To understand this, observe that since $u$ does not depend on $h$, if $u$ is smooth, it has small high frequency oscillations, and hence, for $\xi\neq 0$, and hence points where $u$ is smooth and $\xi\neq 0$ do not appear in $WF_h(u)$. However, where $u$ is not smooth, $u$ has high frequency oscillations and so these points do appear.
To see that $WF_h(u)\cap \mathbb{R}^d\times {0}=$ supp $(u)\times{0}$ observe that $$\mathcal{F}_h(\chi u)(0)=\langle u, \chi\rangle \neq 0$$ for $\chi$ supported near $x_0\in$ supp $(u)$ and $$\mathcal{F}_h(\chi u)(0)=0$$ for $\chi$ supported near $x_0\notin$ supp $(u)$.
To see the second part of $WF_h(u)$, simply use the characterization of $wF(u)$ WF(u)$that for$\xi\neq 0$,$(x_0,\xi_0)\in (x_0,\xi_0)\notin WF(u)$if and only if for$\chi$supported sufficiently close to$x_0$, all$N>0$and$\xi$in a conic neighborhood of$\xi_0$, $$|\mathcal{F}(\chi u)(\xi)|\leq C_N\langle \xi \rangle ^{-N}.$$ Thus,$(x_0,\xi_0)\in (x_0,\xi_0)\notin WF(u)$if and only if for all$N>0$and all$\xi$in a neighborhood of$\xi_0$, $$|\mathcal{F}_h(\chi u)(\xi)|\leq C_N\langle \xi/h\rangle^{-N}.$$ But, since$\xi_0\neq 0$, this gives$(x_0,\xi_0)\in (x_0,\xi_0)\notin WF(u)$if and only if for all$N>0$and$\xi$in a neighborhood of$\xi_0$, $$|\mathcal{F}_h(\chi u)(\xi)|\leq C_Nh^N$$ and hence if and only if$(x_0,\xi_0)\in (x_0,\xi_0)\notin WF_h(u)$. 1 Heuristically, this result follows because the semiclassical wavefront set measures oscillations at frequency$\frac{\xi}{h}$. To understand this, observe that since$u$does not depend on$h$, if$u$is smooth, it has small high frequency oscillations, and hence, for$\xi\neq 0$, and hence points where$u$is smooth and$\xi\neq 0$do not appear in$WF_h(u)$. However, where$u$is not smooth,$u$has high frequency oscillations and so these points do appear. Below you will find a proof of the statement. To see that$WF_h(u)\cap \mathbb{R}^d\times {0}=$supp$(u)\times{0}$observe that $$\mathcal{F}_h(\chi u)(0)=\langle u, \chi\rangle \neq 0$$ for$\chi$supported near$x_0\in $supp$(u)$and $$\mathcal{F}_h(\chi u)(0)=0$$ for$\chi$supported near$x_0\notin $supp$(u)$. To see the second part of$WF_h(u)$, simply use the characterization of$wF(u)$that for$\xi\neq 0$,$(x_0,\xi_0)\in WF(u)$if and only if for$\chi$supported sufficiently close to$x_0$, all$N>0$and$\xi$in a conic neighborhood of$\xi_0$, $$|\mathcal{F}(\chi u)(\xi)|\leq C_N\langle \xi \rangle ^{-N}.$$ Thus,$(x_0,\xi_0)\in WF(u)$if and only if for all$N>0$and all$\xi$in a neighborhood of$\xi_0$, $$|\mathcal{F}_h(\chi u)(\xi)|\leq C_N\langle \xi/h\rangle^{-N}.$$ But, since$\xi_0\neq 0$, this gives$(x_0,\xi_0)\in WF(u)$if and only if for all$N>0$and$\xi$in a neighborhood of$\xi_0$, $$|\mathcal{F}_h(\chi u)(\xi)|\leq C_Nh^N$$ and hence if and only if$(x_0,\xi_0)\in WF_h(u)\$.