How to remove header on the page with big picture (or a page containing floats)?

Assuming that you use the rotating package, i.e., that sidewaysfigure is a floating environment, fancyhdr's \iffloatpage macro should do the trick.

\documentclass{book}

\usepackage{rotating}

\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhead[LO,RE]{\iffloatpage{}{(The Title)}}
\fancyhead[RO,LE]{\iffloatpage{}{\thepage}}
\fancyfoot{}
\renewcommand{\headrulewidth}{\iffloatpage{0pt}{0.4pt}}

\usepackage{blindtext}

\begin{document}

\blindtext[8]

\begin{sidewaysfigure}
\centering
\rule{1cm}{1cm}
\caption{A rotated figure}
\end{sidewaysfigure}

\end{document}

(The blindtext package is only used to add some dummy text to the example.)

EDIT: In your subsequently delivered MWE, replace

\rhead{\small\TheAuthor}

...

\lhead{\small\normalfont\TheTitle}
\fancyfoot{} 
\fancyhead[RO,LE]{\thepage}

with

\rhead{\iffloatpage{}{\small\TheAuthor}}

...

\lhead{\iffloatpage{}{\small\normalfont\TheTitle}}
\fancyfoot{} 
\fancyhead[RO,LE]{\iffloatpage{}{\thepage}}
\renewcommand{\headrulewidth}{\iffloatpage{0pt}{0.4pt}}