Where did $\mathcal{M}^\mu(k) = \int \mathrm{d^4}x \; \exp(\mathrm{i}k \cdot x)\langle f | j^\mu(x) | i\rangle$, in Peskin and Schroeder, come from? On page 160 Peskin & Schroeder, they say: 

Therefore we expect $\mathcal{M}^\mu(k)$ to be given by a matrix element of the Heisenberg field $j^\mu$: $$\mathcal{M}^\mu(k) = \int \mathrm{d^4}x \; \exp(\mathrm{i}k \cdot x)\langle f | j^\mu(x) | i\rangle.$$

Why do they expect that? 
Where does this come from? I've seen similar expressions with e.g. two Lorentz indices and so on. I've been taught (obviously not enough) qft through path integrals and have a hard time to make connections between the different formalisms. 
 A: this is not a complete derivation of the Ward identity but only an intuitive argument. As is stated it comes from the observation that in a gauge theory with minimal coupling the gauge bosons couple to the fermion current linearly.
So the only way to connect an external photon line is 1) either to an fermion loop or 2) to an current that runs all the way from initial to final external lines. One can convince oneself that a fermion line that starts as an external leg can never terminate within the diagram but must always also exit the diagram.Later in the book, these two cases are discussed separately when the Ward identity is proven explicitly.
The connection to this type of argument is actually a little clearer in the path integral formalism via the Schwinger Dyson Equations. This states that the classical equations of motions are fulfilled up to contact terms in correlation functions.
$$\langle0|\frac{\delta S}{\delta \phi}\text{..other fields..}|0\rangle = \text{contact terms}$$
The variation of the action for the $A_\mu$ field is found by varying the Lagrangian density with respect to $A_\mu$
$$\mathcal{L} = A_\mu(g^{\mu\nu}\partial^2 - \partial^\mu\partial^\nu)A_\nu + j^\mu A_\mu$$
$$\frac{\delta\mathcal{L}}{\delta A_\mu} = (g^{\mu\nu}\partial^2 - \partial^\mu\partial^\nu)A_\nu + j^\mu \overset{\text{Lorenz gauge}}{=} \partial^2 A^\mu + j^\mu$$
Therefore we can express correlation functions of (derivatives of) $A_\mu$ by correlation functions of $j^\mu$ plus contact terms
$$-\partial^2\langle0|TA^\mu\text{..other fields..}|0\rangle = \langle0|Tj^\mu\text{..other fields..}|0\rangle + \text{contact terms}$$
The LHS is exactly what we need to get the residue of the correlation function and thus the contirbution to the transition amplitude. Therefore, this is the place where the off-the-cuff remark by P&S is realized that the rest Matrix element aside from the external photon is  essentially goverend by a correlation function of the current plus the remaining external fields. The contact terms do not have the right singularity structure to contribute to the matrix element. 
$$\begin{aligned}\langle f|i\rangle &= i\epsilon_\mu(k) \int \mathrm{d}^4x e^{-ikx}(-\partial^2)\langle 0|A^\mu\text{..other fields..)}|0\rangle \\&=  i\epsilon_\mu(k) \int \mathrm{d}^4x e^{-ikx}(\langle 0|j^\mu\text{..other fields..)}|0\rangle + \text{contact terms})\end{aligned}$$
