What axiom of ZFC implies that "sets have no repeated elements"?

It is indeed the extensionality axiom that is at play here.

We have

$$\forall x (x \in A \iff x \in B)$$ where $A = \{a,a\}$ and $B=\{a\}$ as for both sets $A$ and $B$, $x$ belongs to one of those set if and only if $x=a$.

Therefore $A=B$ by entensionality.