It's been a while since I've thought about this stuff so take this with a grain of salt. Also, I'm only familiar with this in the context of a finite dimensional phase space, whereas the phase space is infinite-dimensional in the example you give and I'm not sure what extra subtleties that introduces. But the finite-dimensional case might still provide some useful intuition.

Those two caveats aside, I hope the following is of some use:

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). We also have a Poisson bracket operation {A, B} between scalar fields A and B. {A, B} is the Lie derivative of B along the tangent vector field corresponding to the exterior derivative of A (obtained using the correspondence between 1-forms and tangent vector fields provided by the symplectic form).

A continuous symmetry is a flow that preserves the Hamiltonian H and the symplectic form. To a continuous symmetry, we can associate a unique tangent vector field that generates it, which corresponds to a 1-form (which happens to be closed). We can then integrate that 1-form to get a scalar field, which I will call S. We have that {S, H} = 0, this basically says that the vector field corresponding to S generates a symmetry. But the Poisson bracket is anticommutative, so {H, S} = 0, implying that S is a conserved quantity (since the vector field corresponding to H generates the time-evolution flow). Therefore, continuous symmetries correspond to conserved quantities.