Here's a more detailed answer:
The above-mentioned link (recreated below [1]) constructs a recursive relation $E$ on $\omega$, such that $(\omega, E)$ is isomorphic to $(\epsilon_0, \in )$. Then, induction up to $\epsilon_0$ is interpreted as $E$-induction, that is, for every predicate $\phi$, if $(\forall x E y \phi(x))\rightarrow \phi(y)$ then $\forall y \phi(y)$.
[1] Normal derivability and first-order arithmetic. P. Tosi Notre Dame J. Formal Logic 21(2): 449-466 (April 1980). DOI: 10.1305/ndjfl/1093883058 https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-21/issue-2/Normal-derivability-and-first-order-arithmetic/10.1305/ndjfl/1093883058.full