The path integral and Feynman diagrams This question is somewhat of a historical one, but it also contains some physics. I am curious to find how exactly the concept of Feynman diagrams arose (I assume from Feynman's path integral)? 
The leap from path integrals to diagrammatic computations isn't obvious (to me, at least); I'd like to understand better how Feynman's thinking approximately developed. For instance, how did he come up with interpreting the propagator as the propagation of particles? Was there a particular analogy that can be made? Is there any understanding to be gained by learning how the technique was originally developed?
I realize that my phrasing might be a quite vague. If the question is too broad as of right now, please let me know how I can improve on it!
 A: 
For instance, how did he come up with interpreting the propagator as the propagation of particles?

The path integral is usually introduced as a matrix element of the time evolution operator
$$
\langle x_f\lvert\mathrm e^{-\frac{\mathrm i}{\hbar}\hat{H}(t_f-t_i)}\lvert x_i\rangle,
$$
which is a measure of the probability of finding a system in final state and time $x_f,t_f$ when it had been in state $x_i$ at time $t_i$ initially. It is quite plausible to name it propagator as it gives immediate access to the probability that a system, maybe only a single particle, propagates from state $x_i$ to $x_f$ in time $t_f-t_i$. Probably it is more difficult to understand that this notion is still maintained when the path integral is used to calculate the grand partition sum in quantum statistics.

Is there any understanding to be gained by learning how the technique was originally developed?

The idea of symbolizing formulae by nodes and connections between them is used in many other fields and was probably not new at the time. The idea is basically that of an isomorphism between a class of graphs and, given an unambiguous translation rule, the formulae at hand. This gives intuitive connection to graph theory and eases its application, for instance when a diagram is called 'connected' or 'disconnected', meaning that the respective formula can be factorized or not.
Another example of this kind that is not related to Feynman is the diagrammatic treatment of the classical Ising model.
A: I was taught that the Feynman diagrams arose as a smart way to write down the intricate computations appearing in the perturbative approach to the path integral.
The keystone is the well known Wick rule, that allows one to compute standard and Grassmannian integrals of correlations with Gaussian measure, e.g. an expression like 
$$\int dx_1\cdots dx_n \ x_{j1}\cdots x_{jn} \ \exp\{- (\hat{x},A\hat{x})\} $$
is rewritten as a sum of several terms, one for each way of "contracting" all the $x_{j1}\cdots x_{jn}$ into couples. In particular each couple of $x$'s contracted, will give also a contribute proportional to an entry of the inverse of $A$.
In the path integral formulation applied to QFT we will need to compute similar integrals where the $x$ are replaced with fields, and the $A$ of the quadratic term is a less trivial object, but is assumed the Wick rule is still true. (At least, so I was taught.)
The inverse of $A$ need a suitable generalization, and it is taken to be its Green function, so you see the Wick rule will make propagators appear.
In order to describe interacting theories you need additional terms in the exponential argument, like $(\hat{J},\hat{x})$ or $\hat{x}{}^4$. This ruins the game since now the Wick rule does not apply anymore. Here enters the idea of expanding the exponential of the new terms $(\exp\{f(x)\} = 1 + f(x) + f(x)^2/2 + \cdots)$, so that you find yourself with a series of integrals computable via the Wick rule. Depending on the field type, bosonic (standard integral) or fermionic (Grassmann integral), and on the terms you have put in the exponential, you can represent the contractions of the Wick rule in a pictorial way respecting some set of rules (the Feynman rules), of course the obtained drawings are the Feynman diagrams.
In general you will have vertices for each field appearing in the integral (outside the exponential of the Gaussian measure) and the contractions among couples will be represented by lines.
A reference I find very interesting is "Non-perturbative renormalization" by Vieri Mastropietro, in the section "Grassmannian measures" the Feynman diagrams are presented as a very natural way to represent the Wick rule for the Grassmannian integral, without mentioning anything about QFT.
A: I read in either one of Feynman's books or in the biography Genius: The Life and Science of Richard Feynman by James Gleick, that Feynman was at a conference in a hotel room trying to work out some path integral, in his pajamas, and at some point found himself surrounded by a bunch of pieces of paper, each containing a term in a perturbation expansion. These basically were so-called Feynman diagrams. If I remember correctly (I don't have the book in front of me) he met with someone else at the conference who had used a similar idea and they realized that it was a good idea and shared it with others.
P.S. That book by Gleick is really good.
EDIT: According to the comments below, the story I'm remembering comes from The Pleasure of Finding Things Out. Furthermore, the actual explanation of where the diagrams come from appears in the book a few pages before the story I described.
