The simplest way I've seen to explain Noether's theorem comes from Feynman's Messenger Lectures.

Suppose -- as we have in modern physics -- that the laws of physics can be phrased in a certain way. Namely:

(1) There is some number we can calculate for all of the possible paths that an object could take from point A to point B. We call this number the "action" of the path.

(2) Of all of the paths that an object can take, the one which it does has a stationary action -- meaning that if you wiggle the path a little bit, the action remains the same. This can happen for example if the action is at a minimum for that path -- if you're at a minimum then you can't go any lower, and so there cannot be a proportionate response to any slight change in path.

This work was already done before Noether -- the observation, I mean, that there is a definition of "action" for all known laws in physics such that you can phrase those laws as a stationary-action principle. An example is light. All of these complicated rules of lenses and light bending when it enters water can be phrased as the much simpler rule that "light always takes the least-time path." The action of the path is then the time it takes for light to travel along that path from A to B.

Now, suppose the action has a continuous symmetry. For example, it might be the case that if A and B were moved a centimeter to the right, to two new points A' and B', then the actions on the corresponding paths between them would remain the same.

Now construct a new path: We go from A to B like so: we go from A to A', follow the physical path to B', then go to B, rather than following the physical path from A to B.

Call these two paths:

(1) A → B

(2) A : A' → B' : B

You have two competing facts. First off:

(1) the symmetry: action(A' → B') = action(A → B)

(2) stationary action: action(A : A' → B' : B) = action(A → B).

In other words, as long as the little wiggles described by ":" are very small, the stationary action principle holds and we don't change the action when we move from a physical path.

It must then be the case that action(A : A') = - action(B : B') in some sense. In other words, there is some quantity local to both A and B which is exactly the same -- it is the susceptibility of the action to changes if you shifted in one direction or another. The big symmetry also forces some little number to be constant while this thing is taking whatever path it takes.

Whenever you have a stationary-action principle -- which we have for gravity and electromagnetism -- then Noether's principle holds and symmetries in the action become conserved quantities of the physics.

We can also calculate some of these susceptibilities for the most common action principles: we find that symmetry with respect to space-translation (moving 1cm to the side) conserves momentum in that direction. We find that symmetry with respect to rotations about a point conserves angular momentum about that point. We find that if the action remains the same from from one second to another, then the energy is conserved -- literally if someone tells you they've got a "free energy device," they're saying "the laws of physics that this thing uses won't be the same tomorrow as they are today."

But it can get even more interesting. For example, wavefunctions have a phase, but shifting the phase does not change any observables. This shift turns out to have relativistic consequences, and can be associated with conservation of charge. (That's as I understand it, but I only saw a half-sketch of the proof and I haven't worked it out myself, as I have for these other examples.)