For $(x_j)_{j\in\Bbb N}\subset I$ and $B\in\mathcal B((\Bbb R^n)^{\Bbb N})$, define $S((x_j)_{j\in\Bbb N},B):=\{f\in X^I,(f(x_j))_{j\in\Bbb N}\in B\}$. The collection of sets of this form contains the $\sigma$-algebra generated by the semi-norms $f\mapsto |f(x)|$, $x\in I$, which is the same as the $\sigma$-algebra generated by the evaluation maps $f\mapsto f(x)$.
We now show that the collection $\mathcal C=\{S(\mathbf x,B), \mathbf x\in\Bbb R^{\infty},B\in \Bbb R^{\infty}\}$ is a $\sigma$-algebra. The main difficulty is to show that this collection is stable for countable unions. Let $\tau\colon\Bbb N^2\to\Bbb N$ be a bijection, and $(S(\mathbf x_{k,\cdot},B_k))_{k\in\Bbb N}$ elements of $\mathcal C$. Then take $y_k:=x_{\tau^{-1}(k)}$ and $B\in\mathcal B((\Bbb R^n)^{\Bbb N}$ such that $B_k=\pi_{\{k\}\times\Bbb N}^{-1}(B)$, where $\pi_I((x_n)_{n\in\Bbb N})=(x_n,n\in I)$.
The subset of continuous functions cannot be expressed in this form, because we don't control the behavior of a function of $S((x_j)_{j\in\Bbb N},B)$ outside an uncountable set.