Set longtable to fit the page-width

Use the tabu with its environment

\begin{longtabu} to \textwidth {|X|X|X|}
…
\end{longtabu}

For more information, see the documentation of tabu.

\begin{center}
\begin{longtable}{|c|c|c|}

the center environment has no affect on longtable as the table rows are always full width.

The longtable documentation does give an example of how to produce a full width table:

\setlength\LTleft{0pt}
\setlength\LTright{0pt}
\begin{longtable}{@{\extracolsep{\fill}}|c|c|c|@{}}

...

an ancient solution is possible with the package ltablex: Here is a MWE with booktabs, which provides better rules then \hline (I recommend to read at least chapter 2 from texdoc booktabs)

\documentclass{article}
\usepackage{ltablex,booktabs}
\begin{document}
\begin{tabularx}{\textwidth}{XXX}
\textbf{Type of movement in the overlap zone} & \textbf{State of the object in camera 1} & \textbf{State of the object in camera 2}  \\ \toprule
\endhead
\caption{Table title.}
\endfoot
Object moving from camera 1 to camera 2 & Object will eventually be in the \emph{exiting} state and the be \emph{deleted} 
& 
Object will be recognised as a \emph{new} object, then become \emph{to be classified} and eventually \emph{classified}
\end{tabularx}
\end{document}