Connection between conserved charge and the generator of a symmetry

Consider an element $g$ of the symmetry group. Say $g$ is represented by a unitary operator on the Hilbertspace $$ T_g = \exp(tX) $$ with generator $X$ and some parameter $t$. It acts on an operator $\phi(y)$ by conjugation $$ (g\cdot\phi)(y) = T_g^{-1}\phi(y) T_g = e^{-tX}\phi(y) e^{tX} = \big[ 1 + t[X,\cdot]+\mathcal{O}(t^2)\big]\phi(y)$$ On the other hand the variation of $\phi$ is defined as the first order contribution under the group action, e.g $$ g\cdot\phi = \phi + \frac{\delta \phi}{\delta t}t+\mathcal{O}(t^2) $$ Since in physics we like generators to be hermitian, rather than anti-hermitian one sends $X\mapsto iX$ and establishes $$ [X,\phi] = -i\frac{\delta \phi}{\delta t} $$

Also, this answer and links therein ought to help you further.

I would like to make an addition to Nephente's answer, because you asked this in your comment, and I also think this is also somewhat part of the full picture here.

Why does the group element act on an operator $\phi$, by conjugation?

This is by no means a mathematically strict answer, but still can be made one.
Consider our $\phi$ acts on a state $|\varphi\rangle$. $$ |\psi\rangle = \phi|\varphi\rangle. $$ Let's state, that our symmetry operator is represented by the following, which is the same operation on every state.
$$ |\varphi'\rangle = T_g^{-1}|\varphi\rangle,\\ |\psi'\rangle = T_g^{-1}|\psi\rangle. $$ From this, we can deduce (by inserting $\mathbb{1} = T_g\ T_g^{-1}$), that $$ |\psi'\rangle = \underbrace{T_g^{-1} \phi\ T_g\ }_{A}\underbrace{T_g^{-1} |\varphi\rangle}_{|\varphi'\rangle}. $$ We can see from this, that $A$ is how we expect the transformed $\phi$ to behave for $|\varphi'\rangle$, $|\psi'\rangle$. Because this is true for $\forall|\varphi\rangle$, $\forall g$ for a given $\phi$, we can conclude, that $$ \phi' = T_g^{-1}\phi\ T_g. $$

(Note, that I think when Nephante wrote, that $T_g$ is how the symmetry operator is represented on the Hilbert space, he really meant it is $T_g^{-1}$, because he then later states, that operators transform by $T_g^{-1}\phi\ T_g$.)