Putting a matrix inside of caption gives \caption has an extra }

\begin{matrix} and \end{matrix} are fragile commands that do not survive the writing process being connected to \caption -- the content is written to the .aux file and to the .lof file later on with \@writefile.

In order to prevent the fragility there, the commands \begin{matrix} and \end{matrix} have to be protected with \protect, i.e. \protect\begin{matrix} and \protect\end{matrix}.

If the protection is not activated, \caption prematurely encounters a } which is not the ending bracket of its mandatory argument.

A better way is to apply the optional argument of \caption and prevent the writing of math content to the .aux and .lof file, i.e. use the short and the long caption style.

However, if fragile content should be written in the short caption argument, the protection must be enabled again.

Alternatively use robust commands.

\documentclass{article}
\usepackage[utf8]{inputenc}

\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{thmtools}       
\usepackage{graphicx}
\setlength\parindent{0pt}
\usepackage[linesnumbered,ruled]{algorithm2e}
\usepackage{hyperref}
\usepackage{caption} 
\usepackage{cleveref}

\begin{document}
\listoffigures
\section{Introduction}
\begin{figure}[ht]
  \centering

  \includegraphics{ente}
  \caption{$x^\star =\protect\begin{bmatrix} \alpha, \beta, \gamma \protect\end{bmatrix}$}
  \caption[Foo content]{$x^\star =\begin{bmatrix} \alpha, \beta, \gamma \end{bmatrix}$}
  \label{fig:just_picture_of_cute_cat}
\end{figure}
\end{document}

With a cat :-P

Add \protect on entering and leaving the environment:

\documentclass{article}
\usepackage[utf8]{inputenc}

\usepackage{amsthm}
\usepackage{amsmath}
\usepackage{graphicx}
\setlength\parindent{0pt}
\usepackage{caption}
\usepackage{hyperref}
\usepackage{cleveref}

\begin{document}

\begin{figure}[ht]
\centering
        \includegraphics[scale=0.5]{vent_cat}
    \caption{$x^\star =\protect\begin{bmatrix} \alpha, \beta, \gamma \protect\end{bmatrix}$}
    \label{fig:just_picture_of_cute_cat}
\end{figure}

\end{document}