How is Hypergraph Isomorphism (HI) reduced to Graph Isomorphism (GI) in polynomial time?
A description is given in the first paragraph of [1], where HI stands for Hypergraph Isomorphism and GI for Graph Isomorphism:

Given a pair of hypergraphs $X=(V,E)$ and $X'=(V',E')$ as instance for HI, the reduced instance of GI consists of two corresponding bipartite graphs $Y$ and $Y'$ defined as follows. The graph $Y$ has vertex set $V \uplus E$ and edge set $E(Y) = \{\{v,e\} \mid v \in V, e \in E, v \in e\}$, and $Y'$ is defined similarly. Here, $C \uplus D$ denotes the disjoint union of the sets $C$ and $D$. It is easy to verify that $Y \simeq Y'$ if and only if $X \simeq X'$ assuming that $V$ can be mapped only to $V'$ and $E$ can be mapped only to $E'$. This latter condition is easy to enforce.

The other direction is trivial as every graph is also a hypergraph.

[1] Arvind, Vikraman, Bireswar Das, Johannes Köbler, and Seinosuke Toda. &quot;Colored hypergraph isomorphism is fixed parameter tractable.&quot; Algorithmica 71, no. 1 (2015): 120-138.