You’re right about the typo in the third to last line: that should be $U_i \in \mathcal{T}_i$.
For your first question, remember how the product topology is defined: the sets of the form $\bigcap\{p_i^{-1}[U_i]:i \in F\}$, where $F$ is a finite subset of $I$ and each $U_i \in \mathcal{T}_i$, are a base for the product topology. This means that every open set in the product is a union of sets of this form. Since $x \in U$, and $U$ is open in the product, $U$ is a union of sets of this form, and one of them must contain $x$. Note that this implies that $x_i \in U_i$ for each $i \in F$.
For your second question, remember that $x_i$ was chosen so that $p_i[[\mathcal{U}]] \to x_i$, so every open nbhd of $x_i$ is in $p_i[[\mathcal{U}]]$. In particular, $U_i$ is an open nbhd of $x_i$, so $U_i \in p_i[[\mathcal{U}]]$. This means that there is some $V \in \mathcal{U}$ such that $p_i[V] = U_i$. But then $p_i^{-1}[U_i] \supseteq V \in \mathcal{U}$, so $p_i^{-1}[U_i] \in \mathcal{U}$ (because $\mathcal{U}$ is an ultrafilter).