First you need to enter the starting and ending points of the arrows (@{x}) in the molecule, then use \chemmove to define the arrows.
\documentclass[a4,10pt]{article}
\usepackage{chemmacros}
\usepackage{chemfig}
\usepackage{tikz}
\begin{document}
\chemleft[
\chemfig{H(-[2,1.3,,,,draw=none])-@{a}C(-[6]H)=[@{b}]C(-[6]H)-[@{c}]O-[@d]@{e}H}
\chemmove{
\draw[red,shorten <=2pt,shorten >=1pt](b).. controls +(90:8mm) and +(90:8mm)..(a);
\draw[red,shorten <=1pt,shorten >=1pt](e).. controls +(90:18mm) and +(120:18mm)..(a);
\draw[red,shorten <=1pt,shorten >=1pt](d).. controls +(-90:6mm) and +(-90:6mm)..(c);
}
\chemright]
\end{document}
the "invisible bond" on the first hydrogen adds a space above the molecule so that the arrows remain inside the [...]
