What exactly is varied in the left-hand-side of Schwinger's quantum action principle?

Schwinger’s quantum action principle states, that if you "vary the field Operators" $$hat{phi’}(x’) = hat{phi}(x) + delta hat{phi}(x)$$ (which changes the Action Operator), then "the variation of the transition Amplitude" is equal to the the variation of the Action. In the Heisenberg picture, given
$$
x’ = x + delta x(x)
$$
and
$$
hat{phi’}(x’) = hat{phi}(x) + deltahat{ phi}(x)
$$
you can calculate what the variation of the action $delta S$ is supposed to be. Further more, you can give some kind of meaning to these mathematical operations: Consider the state of the field being $| Psi rangle$, then the transformation changes the mean value of the field accordingly to
$$
langlephi’rangle(x’) = langlephirangle(x) + langledelta phirangle(x)
$$
In the same manner, the mean value of the action that "this path" has will change by $delta S$.

So much for the right hand side. Now for the left hand side, where I always read something like:

$$
delta langle phi_1, tau_1| phi_2, tau_2 rangle
$$

Question: What is the meaning of $delta$ in this expression?

To be more specific: Usualy a $delta$ indicates a difference between two values, presumably between $ langle phi_1, tau_1| phi_2, tau_2 rangle$ and $ langle phi_1, tau_1|’ |phi_2, tau_2 rangle’$. Which leads me to the question, how $|phi_2, tau_2 rangle’$ is connected to $|phi_2, tau_2 rangle$. Further more, since there is some transformation mapping $|phi_2, tau_2 rangle$ to $|phi_2, tau_2 rangle’$, how is this transformation connected to the transformation of the operator?

I’m asking this because I’m somehow lacking an intuitive understanding of what exactly I am varying here, and what exactly the principle says, and it seems that nobody ever dared to write it down in it’s whole explicitly.