Small note (i.e. footnotesize) below figure

To have the legend occupy the same width as the associated graph, one may encase both in a minipage environment, as is done in the following MWE (minimum working example). Note the use of \par at the end of the footnote-sized material. Without the \par directive -- or a trailing blank line, which is the functional equivalent of a paragraph break in Tex -- you'd get the smaller font size but the wider line spacing that's appropriate for the document's normal font size.

\documentclass{article}
\usepackage[demo]{graphicx} % leave off '[demo]' in real document
\begin{document}
\hrule
(this rule just demonstrates the width of the textblock\dots)

\begin{figure}[h]
\centering
\begin{minipage}{0.65\textwidth} % choose width suitably
\includegraphics[width=\linewidth]{mygraphicfile}
{\footnotesize Here are some notes that go with the graph. Here are some 
notes that go with the graph. Here are some notes that go with the graph. 
Here are some notes that go with the graph.\par}
\end{minipage}
\caption{Historical shock decomposition, 2007Q1--2012Q4}
\end{figure}
\end{document}

---

Addendum: If you want the explanatory stuff to go after the caption, you could place it in its own separate minipage environment, taking care to give it the same width as the graph. (I know there's no arguing about taste, but to me placing the explanatory stuff that far away from the graph doesn't look right.)

\documentclass{article}
\usepackage[demo]{graphicx} % leave off '[demo]' in real document
\begin{document}
\hrule
(this rule just demonstrates the width of the textblock\dots)

\begin{figure}[h]
\centering
\includegraphics[width=0.65\textwidth]{mygraphicfile}
\caption{Historical shock decomposition, 2007Q1--2012Q4}

\medskip % induce some separation between caption and explanatory material
\begin{minipage}{0.65\textwidth} % choose width suitably
{\footnotesize Here are some notes that go with the graph. Here are some notes that go with the graph. Here are some notes that go with the graph. Here are some notes that go with the graph.\par}
\end{minipage}
\end{figure}
\end{document}

Here even \tiny, not only \footnotesize :-)

\documentclass{article}
\usepackage{graphicx}

\begin{document}

\begin{figure}
\includegraphics[width=2cm]{it}

{\tiny I want to include a small note (i.e. footnotesize) below a figure. How do I do that? If I use caption, the note has a caption size and interferes with my caption.}
\caption{Normal text }
\end{figure}


\end{document}