Well, I don't know about any intuitive explanation besides intuition gained by understanding the underlying math (mainly differential geometry, Hamiltonian mechanics and group theory). So with the risk of not giving you quite what you want, I'll try to approach the problem mathematically.

If you know Hamiltonian mechanics then the statement of the theorem is exceedingly simple. Assume we have a Hamiltonian $H$. To this there is associated a unique Hamiltonian flow (i.e. a one-parameter family of symplectomorphisms -- which is just a fancy name for diffeomorphisms preserving the symplectic structure) $\Phi_H(t)$ on the manifold. From the point of view of Lie theory, the flow is a group action and there exists its generator (which is a vector field) $V_H$ (this can also be obtained from $\omega(\cdot, V_H) = dH$ with $\omega$ being the symplectic form). Now, the completely same stuff can be written for some other function $A$, with generator $V_A$ and flow $\Phi_A(s)$. Think of this $A$ as some conserved quantity and of $\Phi_A(s)$ as a continuous family of symmetries.

Now, starting from Hamiltonian equation ${{\rm d} A \over {\rm d} t} = \left\{A,H\right\}$ we see that if $A$ Poisson-commutes with $H$ it is conserved. Now, this is not the end of the story. From the second paragraph it should be clear that $A$ and $H$ don't differ that much. Actually, what if we swapped them? Then we'd get ${{\rm d} H \over {\rm d} s} = \left\{H,A\right\}$. So we see that $A$ is constant along Hamiltonian flow (i.e. conserved) if and only if $H$ is constant along the symmetry flow (i.e. the physical laws are symmetric).

So much for why the stuff works. Now, how do we get from symmetries to conserved quantities? This actually isn't hard at all but requires some knowledge of differential geometry. Let's start with most simple example.

Translation

This is a symmetry such that $x \to x^\prime = x + a$. You can imagine that we move our coordinates along the $x$ direction. With $a$ being a parameter, this is a symmetry flow. If we differentiate with respect to this parameter, we'll get a vector field. Here it'll be $\partial_x$ (i.e. constant vector field aiming in the direction $x$). Now, what function on the symplectic manifold does it correspond to? Easy, it must be $p$ because by differentiating this we'll get a constant 1-form field $dp$ and then we have to use $\omega$ to get a vector field $\partial_x$.

Other way to see that it must be $p$: suppose you have a wave $\exp(ipx)$. Then $\partial_x \exp(ipx) = ip \exp(ipx)$ so momentum and partial derivatives are morally the same thing. Here we're of course exploiting the similarity between Fourier transform (which connects $x$ and $p$ images) and symplectic structure (which combines $x$ and $p$).

Rotation

Now onto something a bit harder. Suppose we have a flow 
$$\pmatrix{x \cr y} \to \pmatrix{x' \cr y'}=  \pmatrix{\cos(\phi) &amp; \sin(\phi) \cr - \sin(\phi) &amp; \cos(\phi)} \pmatrix {x \cr y} $$
 This is of course a rotational flow. Here we'll get a field $y {\rm d}x - x {\rm d} y$ and the conserved quantity of the form $y p_x - x p_y$ which can in three dimensions be thought of as a third component of angular momentum $L_z$.

Note that the above was done mainly for illustrative purposes as we could have worked in polar coordinates and then it would be actually the same problem as the first one because we'd get the field $\partial_{\phi}$ and conserved quantity $p_{\phi}$ (which is angular momentum).

Can Noether's theorem be understood intuitively?
I am currently musically directing a community theater production of Sweeney Todd.  

We have cast a female to play the part of Toby, and in the song "Not While I'm Around", she is not able to achieve sufficient volume on the opening notes of the number.  

The song starts on an Eb, moves to F, then Eb, Ab, and Bb.  She has a great tenor timber to her voice, and once she gets up to the Ab, she sounds great.  We have 16 days until the show opens, and my question is twofold.  

First, are there any good exercises that a singer can do in order to improve the volume on the low end of their range?  

Second, if it comes close to performance time, what does music theory have to say about these two choices: 


changing the melody to something like Ab, Gb, Ab, Ab, Bb, or 
having her begin the song an octave higher, and then on the third line of the song change back to the original octave?  


I personally do not like how the second choice sounds, as it seems to bring about a very drastic change, but I am wondering if there is precedence for something like this, or if conventional musical theory has anything to say.

Here is the first two bars, which is repeated in the next two bars:

