= Solution
Work first over an affine neighbourhood $\operatorname{Spec}A$ of $y=\mathfrak p$, with an affine chart $\operatorname{Spec}B$ in its inverse image. The <scheme-theoretic fibre> on this chart has ring
$$
B\otimes_A\kappa(\mathfrak p)
\cong S^{-1}B/\mathfrak p S^{-1}B,\qquad S=A\setminus\mathfrak p,
$$
where $\kappa(\mathfrak p)$ is the <residue field>. The <prime ideal correspondence for localization> and the correspondence for a quotient identify its points with primes $\mathfrak q\subset B$ containing $\mathfrak p B$ and disjoint from $S$. These two conditions are exactly $\mathfrak q\cap A=\mathfrak p$.
Under this bijection, each principal open $D(b\otimes1)$ corresponds to $D(b)\cap f^{-1}\{y\}$. These are bases for the two topologies: every element of the fibre ring is represented by $b/s$, with the image of $s$ invertible. Thus the bijection is a <homeomorphism>. The identifications agree on overlapping charts and glue, proving
$$
\boxed{|X_y|\cong f^{-1}\{y\}\text{ with its subspace topology}.}
$$
The <scheme-theoretic fibre> may nevertheless carry nilpotents, so this topological statement does not identify its structure sheaf with a naive restriction of $\mathcal O_X$.
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