"I think it is fair to recognize at least 3 important topics:
 

Topic 1. The Riemann (Van Mangoldt, etc.) relation between primes and zeros is considered as an identity between two distributions

Comment: It is a little anachronistic to use the term 'distribution', but this is the idea. In fact Weil's aim in this regard seems to have been at first to express 'the most general' Riemann-like relation between primes and zeros. One can argue whether such a thing exists, and more strongly still whether it really has been achieved by Weil. Also at least prior work of Guinand existed at the time. Anyway, with hindsight, Weil notes in the Comments to his Oeuvres that he could as well have restricted himself to the simplest possible (smooth with compact support) test-functions (see Topic 3 below).
 

Topic 2. The discovery that the local term of the 'prime-side' of this equality of distributions, corresponding to a finite prime or to an infinite prime, 'lives' on the local field, which is the completion of the global number field at this place. Furthermore these local contributions are susceptible to being written in quite similar ways, whether the place is finite or infinite.

Comment (more below): This is clearly where the true innovation of the paper resides. In my opinion this is why the explicit formula deserves to be called 'the Riemann-Weil' formula. The usually stated formula (which only reflects Topic 1) has its key ideas in Riemann, and it is not what I am talking about here. What I am talking about here includes a surprising formula for the conductor of a ramified character, which Weil, in relation with a result in Class Field Theory by Tate and Sen, has generalized in his later 1972 paper. He explains this in his book Basic Number Theory (appendix on the Herbrand distribution). In the 1952 paper, this surprising formula appears as a Deus Ex Machina, nothing is explained about its origin, except that it is a simple calculation if one wants to check it (which is indeed simple and elementary, and not spelled out in Weil's paper; it is left for the attentive reader). What is so striking is that this (elementarily-checked) formula suggests how to rewrite the contribution of the infinite places. This, Weil does, with the help of choosing the same 'finite-part' notation for different things, so that in the end everything looks nice and gentle and all alike at all places. Even when dealing with archimedean places, this is subtle: if you have, say, a complex place, you have to show (it is not obvious if you don't know in advance) that a certain distribution on the positive reals actually comes from the non-zero complex numbers. The least one can say is that Weil's treatment of the real and complex places is cumbersome.
 

Topic 3. The Riemann Hypothesis is expressed as a positivity property of the distribution for which two completely distinct expressions have been given in Topics 1 and 2: one expression involves directly the zeros, the other is a distribution on the group of idele classes.

Comment: This last step is completely elementary, and has no arithmetic in it. It deals with the zeros in the most direct manner, and the positivity criterion has only to be checked on smooth with compact support test-functions, as more or less suggested in the Comments to the Oeuvres.
 

And the Finale: Weil mentions that everything could have been done with a function field. The Riemann Hypothesis was proved by himself there, and so the positivity of the distribution holds in that case. He says this is the most convincing argument to believe in the truth of RH for number fields too.
 

Further comments:

on Topic 3:

The positivity criterion can also be expressed in a more minimal manner, as in a paper by Li, (Journal of Number Theory), and the further work by Bombieri and Lagarias "On Li's criterion for the Riemann Hypothesis" (Journal of Number Theory, 1999).
 

on Topic 2:

Shai Haran (paper in Inventiones) discovered that more uniformity could be given to the local terms. For this he interpreted them as additive convolution and obtained a formula which is identical in all cases. He especially focused on the semi-group whose infinitesimal generator corresponds to this local term.

If you forget about the semi-group and just look at the infinitesimal generator you get the distribution which is Fourier Transform of log(|x|) (note that the Weil's 'Finite Parts' expressions then need an additional minus sign, but this is also how they appear when you are counting the zeros positively, and you want to do that in the statement of the positivity criterion.)

So basically, although he chose not to write it this way, Haran observed that the local terms of the Explicit Formula were all just additive convolution with log(|y|) (additive Fourier Transform of log(|x|)). He gives explicitly the computation for: a finite prime, not dividing the conductor, and for the real prime. To be complete, you then need to check for the complex prime, and most importantly for the primes dividing the conductor. It all works OK, so really Haran has improved upon Weil. But we have then four distinct necessary computations (not to mention dealing with the discriminant of the number field.)

And my contribution is this: it is centered on the dilation-invariant operator log(|x|) + log(|y|), instead of just log(|y|). Of course for the Explicit Formula, you evaluate log(|x|) at 1, so this gives 0. Still good reasons exist to keep it. My contribution, as published in my CRAS note from the year 2000, has been to provide a proof which is the same in all cases. This means you do not formulate the Explicit Formula first on the positive reals, then as an afterthought realize it did come from the local fields: you directly jump from the critical line to the local fields."
 

Jean-Francois Burnol,
Mardi 5 Juin 2001