$I$ is generated by the monomials that are products of $n-d+1$ (distinct) indeterminates.
Take $\Delta_m=\{G\subset[1,n]:|G|\le m\}$. This is a simplicial complex on $[1,n]$ whose facets are the subsets $F$ of $[1,n]$ with $|F|=m$. Define $I_{\Delta_m}$ as being the ideal of $K[X_1,\dots,X_n]$ generated by all the monomials $X_{i_1}\cdots X_{i_s}$ with $\{i_1,\dots,i_s\}\notin\Delta_m$. From Bruns and Herzog, Cohen-Macaulay Rings, Theorem 5.1.4, one knows that $I_{\Delta_m}=\bigcap P_F,\;\;\;\; (*)$ where the intersection is taken over all facets $F$ of $\Delta_m$, and $P_F$ denotes the ideal generated by all $X_i$ with $i\notin F$. In our case $\{i_1,\dots,i_s\}\notin\Delta_m$ means that $s\ge m+1$ and then $I_{\Delta_m}$ is generated by all the monomials $X_{i_1}\cdots X_{i_s}$ with $s\ge m+1$. (In fact one can say that $I_{\Delta_m}$ is generated by all the monomials $X_{i_1}\cdots X_{i_s}$ with $s=m+1$.) On the other side, $P_F$ is generated by all $X_i$ with $i\notin F$, that is, $P_F$ is generated by all $X_i$ with $i\in [1,n]-F$. Looking back to $(*)$ we can deduce that the right hand side coincides with $\bigcap P_G$ where $G$ runs over all the subsets of $[1,n]$ with $|G|=n-m$. Now take $m=n-d$ and get the conclusion.
Remark. This is a simpler answer which I've found later: one knows that a finite intersection of monomial ideals is a monomial ideal, therefore $I=\bigcap_{|F|=d} P_F$ is a monomial ideal. A monomial $X_{i_1}^{a_{i_1}}\cdots X_{i_s}^{a_{i_s}}$, denoted by $X_G^a$, where $G=\{i_1,\dots, i_s\}$ and $a=(a_{i_1},\dots,a_{i_s})$, belongs to $P_F$ iff $G\cap F\neq\emptyset$, so $I$ contains all the monomials $X_G^a$ with $G\cap F\neq\emptyset$ for all $F\subset[1,n]$ with $|F|=d$. This shows that $|G|\ge n-d+1$ and get the (same) conclusion.