It's been a while since I've thought about this stuff so take this with a grain of salt. But if my memory serves correctly, it goes something like this:

We can define a canonical symplectic form on phase space in terms of the Lagrangian. The symplectic form gives us a bijective correspondence between tangent vector fields and 1-forms (it works the same as with Riemannian manifolds, the key is just that we have a perfect pairing on tangent spaces). Continuous symmetries are flows on phase space which preserve the symplectic structure. To a continuous symmetry, we can associate a tangent vector field (its derivative, sort of), and therefore a 1-form (via the symplectic form). This 1-form happens to be closed (I don't remember why, might edit this post later if I remember), and therefore we can get a scalar field by integrating it, at least locally. This scalar field is the conserved quantity.

This is all closely related to the Poisson bracket. With the Poisson bracket, any scalar field S can also be thought of as a mapping from scalar fields to scalar fields (the [S, _] operator). This mapping satisfies the Leibniz rule, and is just the derivative operator corresponding to some tangent vector field. Any continuous symmetry (which is a particular type of tangent vector field) can be obtained this way, and the conservation law is just a special case of the Poisson bracket identity [S, S] = 0.