The easiest way to finish a page (and create a new one) is the \newpage command. This will add enough vertical space to fill the current page, and then add a page break.
\documentclass{article}
\title{Appendix A: User Manual}
\author{
John Smith\\University XyZ\\Albert Square\\Walford\\NC1 8AE
}
\begin{document}
\maketitle
\newpage
\tableofcontents
\newpage
\section{Introduction}
Lorem Ipsum.
\end{document}
But in your case you will have an almost empty first page - in an article you normally don't want a separate title page. Simply put the table of contents (\tableofcontents) on the same page, and then start with the first section:
\documentclass{article}
\title{Appendix A: User Manual}
\author{
John Smith\\University XyZ\\Albert Square\\Walford\\NC1 8AE
}
\begin{document}
\maketitle
\tableofcontents
\section{Introduction}
Lorem Ipsum.
\end{document}
(You will need at least two LaTeX runs to see the full table of contents.)