[funcantor]

I guess very roughly you can say that if you describe a system with some set of parameters, which when changed in particular ways leave the system's quantities (energy, momentum, etc..) unchanged then you have a connection between symmetry and conserved quantities.

[Koshkin]

But note that this can be misleading (as could be any hand-wavy “explanation”). For example, a time shift does not affect the system’s angular momentum, but that does not mean that the conservation of the latter follows from the symmetry alluded to by the former.