This is Proposition II.6.6 in Hartshorne. We assume that $X$ is a Noetherian, integral, and separated scheme which is regular in codimension 1, i.e. every local ring of dimension one is regular.
For this question, I am solely interested in the Type I points, namely, points $x \in X \times \mathbf{A}^1$ whose closures are of codimension one and under the image of the projection map to $X$, is another point $y$ of codimension one. Hartshorne claims
$x$ is the generic point of $\pi ^{-1} (y)$. Its local ring $\mathcal{O}_x \cong \mathcal{O}_y[t] _{\mathfrak m_y}.$
However, I think there is a mistake here, I believe the correct statement here is $\mathcal{O}_x \cong \mathcal{O}_y[t] _{\mathfrak m_y[t]}$, since we have the following conclusion that could be proved easily.
Let $B$ be a domain and $\mathfrak p$ a prime ideal of $B$, then we have $B[t]_{\mathfrak p[t]}\cong B_\mathfrak p[t]_{\mathfrak pB_{\mathfrak p}[t]}.$
