The Feynman diagrams are a pictorial presentation of individual terms in a specific expansion of a deformation of the free field Hamiltonian. Free quantum fields can be constructed rigorously and we can compute the probability densities we would observe for measurements we might make in a world in which a free quantum field described physics; no interactions is not enough to do any interesting physics of the real world, but at least it's rigorous math.
The deformations we introduce
can be written as a point-dependent unitary transformation ##\hat\xi(x)=\hat U^{-1}(x)\hat\phi(x)\hat U(x)##, where ##\hat\phi(x)## is a free quantum field and ##\hat U(x)## is the time-ordered exponential $$\hat U(x)=\mathrm{T}\left[\mathrm{e}^{-\mathrm{i}\int\limits_{y\preceq x} \hat H_\mathrm{int}(y)\mathrm{d}^4y}\right].$$ The integral is over all points ##y\preceq x## that causally precede ##x##, which deforms the Hamiltonian density of the free field, ##\hat H_0(x)\mapsto\hat H_0(x)+\hat H_\mathrm{int}(y)##. If we work formally, ignoring niceties of whether ##\hat H_\mathrm{int}(y)## exists and whether the integral exists, expansions include integrals that can be pictorially presented as Feynman diagrams.
In the above, I'm channeling Itzykson & Zuber's "Quantum Field Theory", which is the standard textbook from the 1970s and 80s (with the slight change that I've presented it Lorentz covariantly, using ##y\preceq x##,
which has its own interests). I&Z describe how the expansion works in detail. The difference is that back then the path integral had not taken over quite so much, as you see above in the use of a point-dependent unitary deformation. The math is equivalent, but the physical intuition has always seemed to me rather different. If the Peskin & Schroeder is not working for you, one approach would be to try I&Z. If you go there, you'll have to make sure you eventually understand how the P&S world is related to the I&Z world, if you want to talk to people who think in the P&S way, but it will stand you in good stead to know both, not least because you'll be able to talk to people older than about 60 who grew up with I&Z, but also because in my experience quantum computing/quantum optics rarely uses a path-integral approach, so the I&Z approach somewhat better connects to non-HEPhysics. Needless to say, you can try other earlier textbooks as well or instead.
