How can I make underbrace texts have the normal-text size?

\scriptstyle is hardcoded into environment subarray, which is used by \substack. As workaround, \scriptstyle can be locally redefined as \textstyle:

\documentclass{article}
\usepackage{amsmath}

\begin{document}
\[
  \implies \underbrace{x(t) = z(t)}_{%
    \let\scriptstyle\textstyle
    \substack{x(t)=y(t)\\y(t)=z(t)}}
\]
\end{document}

A cleaner result can be achieved by using environment array together with \textstyle. This fixes also the too narrow line spacing of \substack:

\documentclass{article}
\usepackage{amsmath}

\begin{document}
\[
  \implies \underbrace{x(t) = z(t)}_{%
    \textstyle
    \begin{array}{c}
      x(t)=y(t)\\
      y(t)=z(t)
    \end{array}
  }
\]
\end{document}

And a third variant using gathered

\documentclass{article}
\usepackage{amsmath}

\begin{document}
\[
  \implies \underbrace{x(t) = z(t)}_{%
    \textstyle
    \begin{gathered}
      x(t) = y(t)\\
      y(t) = z(t)
    \end{gathered}
  }
\]
\end{document}

Use array:

\documentclass{article}
\usepackage{amsmath}

\newcommand{\bunderbrace}[2]{%
  \begin{array}[t]{@{}c@{}}
  \underbrace{#1}\\
  #2
  \end{array}
}

\begin{document}
\[
\implies
\bunderbrace{x(t) = z(t)}{x(t)=y(t)\\y(t)=z(t)}
\]
\end{document}