Finding a more direct way to reach $\mathbb{E} \left( \sum (X_i - \mu)^2 \right) - \mathbb{E} \left( \sum (X_i - \overline{X})^2 \right) = \sigma^2$ Let $X_i$ be independent random variables, $\forall\,i \in \mathbf{n} \equiv \{0,\dots,n-1\}$, with identical expectation value $\mathbb{E}(X_i)=\mu$, and identical variance $\mathrm{Var}(X_i)=\sigma^2$.1  Also, let $\overline{X}$ be their average, $\frac{1}{n}\sum X_i$ (where the summation for all $i\in \mathbf{n}$ is implicit, a convention I'll use throughout).
It is not hard to show, by direct (if slightly tedious) calculation, that
$$
\textstyle \mathbb{E} \left( \sum (X_i - \mu)^2 \right) - \mathbb{E} \left( \sum (X_i - \overline{X})^2 \right) = \sigma^2
$$
Whenever I arrive at a very "simple" result through a "tedious" derivation, as in this case, I get the strong suspicion that there has to be a more direct, and yet entirely rigorous, reasoning to reach it.  Or rather, a way to view the problem that makes the result immediately "obvious".2
In this case, the best I have found goes something like this: the above result follows  from the fact that, first,
$$
\textstyle \mathbb{E}  \left( \sum (X_i - \mu)^2 \right) = n \sigma^2
$$
and, second, if we subtract $\overline{X}$ instead of $\mu$, we have "lost one degree of freedom", and "therefore"
$$
\textstyle \mathbb{E}  \left( \sum (X_i - \overline{X})^2 \right) = (n - 1) \sigma^2
$$
I find this hand-wavy argument thoroughly unconvincing.  (I doubt that those who propose it would believe it if they didn't already know the result from a more rigorous derivation.)
Is there something better?
EDIT:
Here's an example of the kind of argument I'm looking for.  It still has too many gaps to be satisfactory, but at least it shows a reasoning that does not require pencil and paper: it could be delivered orally, or with crude "marks in the sand" (no algebra), and be readily understood.
First we can see that
$$
\textstyle \mathbb{E} \left( \sum (X_i - \mu)^2 \right) \geq \mathbb{E} \left( \sum (X_i - \overline{X})^2 \right)
$$
Why?  Because, the value of $c$ that minimizes $\sum (X_i - c)^2$ is $c = \overline{X}$ (a fact for which I could give a similarly hand-wavy, not-entirely-watertight argument, though easy to prove by tedious computation), which means that whenever $\overline{X} \neq \mu$, we would have $\sum (X_i - \mu)^2 > \sum (X_i - \overline{X})^2$.
Now, what "drives" $\overline{X}$ away from $\mu$ (so to speak) is $\sigma^2$.  So we can conclude that the difference
$$
\textstyle \mathbb{E} \left( \sum (X_i - \mu)^2 \right) - \mathbb{E} \left( \sum (X_i - \overline{X})^2 \right) \geq 0
$$
should increase monotonically as $\sigma^2$ increases...
Fair enough, but why is the difference exactly $\sigma^2$, and not, say, $\sigma^2/n$, or &ast;gasp&ast; $\pi \sigma^2/n$?  Here my hand-waving begins to run out of steam...  It is suggestive that $\mathbb{E}(\overline{X}) = \mu$ and 
$$\mathrm{Var}(\overline{X}) = \frac{\sigma^2}{n} =
\mathbb{E} \left( ( \overline{X} - \mathbb{E}(\overline{X}))^2 \right) = \mathbb{E} \left( ( \overline{X} - \mu )^2 \right)$$
Therefore it is tempting to surmise that, since, for each $i$, $(X_i - \mu) - (X_i - \overline{X}) = \overline{X} - \mu$, then each term $(X_i - \mu)^2 - (X_i - \overline{X})^2$ would contribute, "on average", $\mathbb{E}\left( ( \overline{X} - \mu )^2 \right) = \sigma^2/n$ to the total difference.  This would require justifying the tantalizingly Pythagorean-looking equality:
$$\mathbb{E} \left( ( X_i - \mu )^2 \right) = \mathbb{E} \left( ( X_i - \overline{X})^2 + ( \overline{X} - \mu )^2 \right)$$
...though I readily concede that this is beginning to look as tedious as any algebraic computation.
(I note that in this argument I did not use the fact that in this case $\mathrm{Var}(\sum X_i)=\sum\mathrm{Var}(X_i)$, which is surely the way forward.)

1 The typically given condition is to say that the $X_i$ are independent and identically distributed, but, AFAICT, the last condition is stronger than necessary.  For that matter, as Dilip Sarwate pointed out, the independence condition is also stronger than needed.  It is sufficient that $\mathrm{Var}(\sum X_i)=\sum\mathrm{Var}(X_i)$.
2 The "scare quotes" around "simple", "tedious", and "obvious" aim to convey the concession that these terms are all, of course, in the eye of the beholder.  So "simplicity" is shorthand for subjective simplicity or perceived simplicity, etc.  Also in the eye of the beholder is how much (perceived) tedium seems too much relative to the (perceived) simplicity.  If the difference $\mathbb{E} \left( \sum (X_i - \mu)^2 \right) - \mathbb{E} \left( \sum (X_i - \overline{X})^2 \right)$ had been, say, $\sigma^2/\sqrt{\pi}$, I would not have perceived the standard algebraic derivation as particularly tedious, because $\sigma^2/\sqrt{\pi}$ does not seem to me particularly simple.
 A: Note that if the $X_i$ are independent,
$$
\begin{align}
\mathrm{Var}\left(\bar{X}\right)
&=\mathrm{Var}\left(\frac1n\sum X_i\right)\\
&=\frac1{n^2}\sum\mathrm{Var}(X_i)\\
&=\frac1{n^2}n\sigma^2\\
&=\frac{\sigma^2}{n}\tag{1}
\end{align}
$$
Simply expand and simplify to get
$$
\begin{align}
&\mathbb{E}\left(\sum\left(X_i-\mu\right)^2\right)-\mathbb{E}\left(\sum\left(X_i-\bar{X}\right)^2\right)\\
&=\mathbb{E}\left(\sum\left(X_i^2-2\mu X_i+\mu^2\right)-\sum\left(X_i^2-2\bar{X}X_i+\bar{X}^2\right)\right)\\
&=\mathbb{E}\left(\sum\left(\mu^2+2\left(\bar{X}-\mu\right)X_i-\bar{X}^2\right)\right)\\
&=\mathbb{E}\left(n\mu^2+2n\left(\bar{X}-\mu\right)\bar{X}-n\bar{X}^2\right)\\
&=n\mathbb{E}\left(\mu^2-2\mu\bar{X}+\bar{X}^2\right)\\
&=n\mathbb{E}\left(\left(\mu-\bar{X}\right)^2\right)\\
&=n\frac{\sigma^2}{n}\tag{2}\\
&=\sigma^2
\end{align}
$$
$(2)$ is just $n$ times $(1)$.
A: 
An intuitive and possibly "completely rigorous" derivation of the result

The result in question follows from an application of the
Pythagorean theorem of plane geometry.
Without loss of generality, assume that $\mu=0$ and consider
a fixed point $\mathbf{x} = (x_1,x_2,\ldots, x_n)$ in $\mathbb R^n$. 
Define $\bar{x}$ as $\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i$.
The set of all points
$(y_1,y_2,\ldots, y_n) \in \mathbb R^n$ that satisfy
$$y_1+y_2+\cdots + y_n = n\bar{x}$$ is a hyperplane $H$ in $\mathbb R^n$
that contains $\mathbf{x}$, and it is easily shown that the
point in $H$ that is closest to the origin $\mathbf 0$ is
$\bar{\mathbf{x}} = (\bar{x},\bar{x}, \ldots, \bar{x})$
and that the straight line through $\mathbf 0$ and $\bar{\mathbf{x}}$
is perpendicular to $H$.  Now, the three points $\mathbf 0$,
$\bar{\mathbf{x}}$, and $\mathbf x$ define a plane, and in this
plane, they are the vertices of a right triangle (with right
angle at $\bar{\mathbf{x}}$).  The Pythagorean theorem of plane
geometry tells us that
$$\sum_{i=1}^n x_i^2 = \sum_{i=1}^n \left(\bar{x}\right)^2
+ \sum_{i=1}^n \left(x_i-\bar{x}\right)^2 
= n\left(\bar{x}\right)^2 + \sum_{i=1}^n \left(x_i-\bar{x}\right)^2$$
or, equivalently,
$$\sum_{i=1}^n x_i^2 - \sum_{i=1}^n \left(x_i-\bar{x}\right)^2
= n\left(\bar{x}\right)^2.$$
Now, the above identity holds for all choices of the $x_i$, and
in particular, it holds for all realizations $(x_1, x_2, \ldots, x_n)$
of the random vector $(X_1, X_2, \ldots, X_n)$, that is,
$$\sum_{i=1}^n X_i^2 - \sum_{i=1}^n \left(X_i-\bar{X}\right)^2
= n\left(\bar{X}\right)^2 ~~\text{with probability } 1.$$
Therefore, assuming all the expectations exist, we have that
$$E\left[\sum_{i=1}^n X_i^2\right] - E\left[\sum_{i=1}^n \left(X_i-\bar{X}\right)^2\right] = nE\left[\left(\bar{X}\right)^2\right].$$
Introducing a common mean $\mu$ for the $X_i$'s merely translates
the origin to $(\mu,\mu, \ldots, \mu)$ giving

$$E\left[\sum_{i=1}^n (X_i-\mu)^2\right] 
- E\left[\sum_{i=1}^n \left(X_i-\bar{X}\right)^2\right] 
= nE\left[\left(\bar{X}-\mu\right)^2\right]
= n\cdot\operatorname{var}\left(\bar{X}\right).$$

Notice that the result holds for all random variables with
common mean $\mu$: we have not made any assumptions about
independence or zero correlation or even about common variance.
Now, for the special
case of uncorrelated random variables with common variance
$\sigma^2$, the right side of the above equality is just
$$n\cdot\operatorname{var}\left(\bar{X}\right)
= n\cdot\operatorname{var}\left(\frac{1}{n}\sum_{i=1}^n X_i\right)
= n \cdot \frac{1}{n^2}\sum_{i=1}^n \operatorname{var}(X_i) = \sigma^2$$
giving

$$E\left[\sum_{i=1}^n (X_i-\mu)^2\right] 
- E\left[\sum_{i=1}^n \left(X_i-\bar{X}\right)^2\right] 
= \sigma^2.$$



(Previous answer: no intuition or geometry, just a simple
derivation)
You already have noted that 
$$E\left[\sum_{i=1}^n (X_i-\mu)^2\right] = \sum_{i=1}^n E[(X_i-\mu)^2]
= n\sigma^2.$$
Since $E\left[\bar{X}\right] = \mu = E[X_i]$, we have that
$Y_i = X-\bar{X}$ is a zero-mean random variable, and so $E\left[\left(X_i-\bar{X}\right)^2\right]$ is the variance of $Y_i$ which gives
$$\begin{align}
E\left[\left(X_i-\bar{X}\right)^2\right]
&= \operatorname{var}(Y_i) = \operatorname{var}\left(X_i-\bar{X}\right)\\
&= \operatorname{var}\left(\frac{n-1}{n}X_i-\sum_{j\neq i}\frac{X_j}{n}\right)
&\scriptstyle{\text{write as a weighted sum of the uncorrelated variables }X_i}\\
&= \left(\frac{(n-1)^2}{n^2}+ (n-1)\frac{1}{n^2}\right)\sigma^2
&\scriptstyle{\text{so that we can use 
the formula}\operatorname{var}\left(\sum_i a_iX_i\right)
= \sum_i a_i^2\operatorname{var}(X_i)}\\
&= \frac{n-1}{n}\sigma^2&\scriptstyle{\text{for 
the variance of a sum of uncorrelated random variables}}
\end{align}$$
leading to 
$$E\left[\sum_{i=1}^n\left(X_i-\bar{X}\right)^2\right] = (n-1)\sigma^2.$$
A: I think you want to do the tedious calculations and then extract the key insight.  And for me, the key insight is that, for each $X_i$:
$$
\textstyle \mathbb{E} \left( X_i  \overline{X} \right) = \frac{\sigma^2}{n} +\mu^2
$$
which does require that $Cov(X_i,X_j)=0$ if $i\ne j$.
Once you have this, then you can see that:
$$
\textstyle  \mathbb{E} \left( \sum (X_i - \overline{X})^2 \right) = \mathbb{E} \left( \sum [(X_i -\mu)+(\mu- \overline{X})]^2 \right)
$$
$$
\textstyle  = \mathbb{E} \left( \sum (X_i -\mu)^2+\sum2(X_i -\mu)(\mu- \overline{X})+\sum(\mu- \overline{X})^2 \right)
$$
$$
\textstyle  = n\sigma^2-2\sigma^2+\sigma^2=(n-1)\sigma^2
$$
which is indeed tedious to compute, but this is where your "degree of freedom" shows up.
