Underbrace under sqrt content

I would recommend Mico's way; if you really want the surd, then here is a solution that takes all the vertical space it needs (not like in wh1t3's answer):

\documentclass{article}
\usepackage{mathtools}

\def\sqrtexplained#1{%
  \begingroup
    \sbox0{$#1$}
    \def\underbrace##1_##2{##1}
    \sbox2{$#1$}
    \dimen0=\wd0 \advance\dimen0-\wd2
    \mathrlap{\sqrt{\phantom{\displaystyle#1}\kern\dimen0 }}
    \hphantom{\sqrt{\vphantom{\displaystyle#1}}}
  \endgroup
  #1}


\begin{document}

\[
\sqrtexplained{%
\underbrace{\frac{1}{\tau}
  \underbrace{\int_{-\infty}^{t}
    \underbrace{p_{X}^{2}(\xi)\cdot e^{\frac{-(t-\xi)}{\tau}}}
    _{\substack{\text{Exponential}\\
                \text{time-weighting}\\
                \text{of $p_{X}^{2}(\xi)$}}
     }
   \,\mathrm{d}\xi}
   _{\text{Exponential integration}}}
_{\substack{\text{Exponential averaging}\\
            =\text{Quasi mean-square}}
 }
}
\]

\end{document}

I've also corrected the input errors you are making: \text{-} for a minus sign is wrong; \left and \right should be used only when they are really needed.

I would suggest that you not use the \sqrt{...} symbolism at all because it adds yet another horizontal bar to an expression that's already very busy-looking. Instead, I suggest you write something like:

\documentclass{article}
\usepackage{concrete,eulervm,mathtools}
\begin{document}
$ \biggl(\,
   \underbrace{\frac{1}{\tau} 
   \underbrace{\int\limits_{-\infty}^{t}
   \underbrace{p_{X}^{\,2} (\xi) \cdot e^{-(t-\xi)/\tau}}
      _{\substack{\text{Exponential time-}\\
                  \text{weighting of $p_{X}^{\,2}(\xi)$}}} 
   \,d\xi}
      _{\text{Exponential integration}}}
      _{\substack{\text{Exponential averaging}\\
                  \text{$=$ ``Quasi mean-square''}}}
  \,\biggr)^{1/2} $
\end{document}

You can use \smash and a \vphantom to make it work.

\documentclass{article}
\usepackage{mathtools}
\begin{document}

\[\sqrt{\vphantom{\int_{\text{-}\infty}^{t}x}\smash{
  \underbrace{
    \frac{1}{\tau}
    \underbrace{
      \int_{\text{-}\infty}^{t}
      \underbrace{
        p_{X}^{\,2}\left(\xi\right)\cdot e^{\frac{\text{-}\left(t\text{-}\xi\right)}{\tau}}
        }_{\substack{\text{Expontial}\\\text{time-weighting}\\\text{of $p_{X}^{\,2}\left(\xi\right)$}}} \,\mathrm{d}\xi
      }_{\text{Exponential integration}}
    }_{\substack{\text{Exponential averaging}\\\text{= “Quasi mean-square”}}}
  }}\]

\end{document}

The output looks like this:

I should note that this does not appear to work in inline math-mode. The size of the vphantom does not correctly reflect the integral sign then. Perhaps someone else can explain why this is the case, since I don't know.

Edit after @egreg's comment, an additional vphantom should be added, to ensure the displaymath gets the height of the braces as well. The modified code would look like this:

\documentclass{article}
\usepackage{mathtools}
\begin{document}

\[\sqrt{\vphantom{\int_{\text{-}\infty}^{t}x}\smash{
  \underbrace{
    \frac{1}{\tau}
    \underbrace{
      \int_{\text{-}\infty}^{t}
      \underbrace{
        p_{X}^{\,2}\left(\xi\right)\cdot e^{\frac{\text{-}\left(t\text{-}\xi\right)}{\tau}}
        }_{\substack{\text{Expontial}\\\text{time-weighting}\\\text{of $p_{X}^{\,2}\left(\xi\right)$}}} \,\mathrm{d}\xi
      }_{\text{Exponential integration}}
    }_{\substack{\text{Exponential averaging}\\\text{= “Quasi mean-square”}}}
  }}
  \vphantom{\underbrace{
    \frac{1}{\tau}
    \underbrace{
      \int_{\text{-}\infty}^{t}
      \underbrace{
        p_{X}^{\,2}\left(\xi\right)\cdot e^{\frac{\text{-}\left(t\text{-}\xi\right)}{\tau}}
        }_{\substack{\text{Expontial}\\\text{time-weighting}\\\text{of $p_{X}^{\,2}\left(\xi\right)$}}} \,\mathrm{d}\xi
      }_{\text{Exponential integration}}
    }_{\substack{\text{Exponential averaging}\\\text{= “Quasi mean-square”}}}
  }
  \]

  A long sentence after it to see if there are problems with overruning.

\end{document}

And its output like this: