elsarticle defines \thesection as \appendixname \Alph{section} within \appendix -- as such, \thesubsection expands to \appendix \Alph{section}.\arabic{subsection}, finally being Appendix A.1 etc.
While this might alright for the typesetting, the references might become tedious and lengthy.
Redefinition of \p@subsection by gobbling the content first and re-injection of the pure \Alph{section}.\arabic{subsection} will remove this for the references.
See https://tex.stackexchange.com/a/397724/31729 for a similar approach.
Another possibility is to refine \thesection for the appendix.
\documentclass{elsarticle}
\makeatletter
\def\@gobbleappendixname#1\csname thesubsection\endcsname{\Alph{section}.\arabic{subsection}}
\renewcommand{\p@subsection}{\@gobbleappendixname}
\makeatother
\begin{document}
... briefly explained in~\ref{sec:subAppendix}
\appendix
\section{Appendix}
\subsection{Sub-appendix}
\label{sec:subAppendix}
\end{document}

Regarding the complaints of the OP:
\documentclass{elsarticle}
\makeatletter
\def\@gobbleappendixname#1\csname thesubsection\endcsname{\Alph{section}.\arabic{subsection}}
\g@addto@macro{\appendix}{\renewcommand{\p@subsection}{\@gobbleappendixname}}
\makeatother
\begin{document}
\section{Foo} \label{firstsection}
\subsection{Foo subsection} \label{foosubsection}
... briefly explained in~\ref{sec:subAppendix} or in \ref{firstsection} or in \ref{firstappendixsection} or in \ref{foosubsection}
\appendix
\section{Appendix} \label{firstappendixsection}
\subsection{Sub-appendix}
\label{sec:subAppendix}
\end{document}