Just one clarification to Guillaume's answer: Yes, a model of ZFC is a set
$E$ together with a binary relation $R$ on $E$ such that $(E,R)$ satisfies ZFC.
The relation $R$ is extensional, that is every $x\in E$ is determined by the collection of $y\in E$ with $yRx$, because $(E,R)$ satisfies the axiom of extensionality.
However, even though $(E,R)$ satisfies the axiom of regularity, the relation $R$ need not be well-founded. $(E,R)$ thinks that $R$ is well-founded, but the real world knows
an infinite $R$-decreasing chain in $E$. In this situation, $(E,R)$ is a non-standard model
of ZFC which is clearly not isomorphic to a transitive model with the $\in$-relation.
However, an inaccessible cardinal $\kappa$ actually gives you a transitive model of ZFC,
and this is more than just having any model of ZFc.
Now, it is not the case that the inaccessibility of $\kappa$ somehow miraculously gives you a new set that is a model of ZFC. It is more the case that $\kappa$ is so large that if you cut off the universe at $\kappa$, this initial part of the real universe already satisfies ZFC.
I.e., from the point of view of $V_\kappa$, $\kappa$ already looks like the class of all ordinals.