These images are borrowed from the video → Gely
There exist two half-reactions of hydrogen peroxide:
(1) $H_2O_2$ $+$ $2e^-$ $+$ $2H^+$ $\longrightarrow$ $2H_2O$
(Product: Water)
The oxidation numbers of $O$ vary from $2\cdot(-1)$ $=$ $-2$ to $2\cdot(-2)$ $=$ $-4$,
so we have a reduction with gain of $2e^-$:
(2) $H_2O_2$ $-$ $2e^-$ $\longrightarrow$ $O_2$ $+$ $2H^+$
(Product: The observed oxygen!)
The oxidation numbers of $O$ vary from $2\cdot(-1)$ $=$ $-2$ to $2\cdot0$ $=$ $0$,
so we have an oxidation with loss of $2e^-$:
The $Fe^{3+}$ ion liberated by iron(III) chloride in water serves as catalyst, i.e. it accelerates the spontaneous reaction!
The complete equation:
$H_2O_2$ $+$ $2e^-$ $+$ $2H^+$ $\longrightarrow$ $2H_2O$
$H_2O_2$ $-$ $2e^-$ $\longrightarrow$ $O_2$ $+$ $2H^+$
$2H_2O_2$ $\longrightarrow$ $2H_2O$ $+$ $O_2$
As hydrogen peroxide is simultaneously reduced and oxidized, it is said to dismutate .