There are two explanations in Silverman ( Arithmetic of Elliptic Curves), one in exercises developing the Weil reciprocity law ( for algebraic curves) and then generalizing, and then there is a different, somewhat computational (in my opinion) proof in one of the chapters. 


[I should also point out that in case of elliptic curves over complex numbers there is a rather simple description of this pairing in terms of determinant of a matrix!, see Ribet Stein : Hecke Operators ... for example] 


While I do understand the proofs, I have heard that there is a conceptual explanation of this pairing?


Is there a uniform construction for the Hilbert Symbols in number field case, which is again a "form" of Weil pairing? 



Conceptualizing Weil Pairing for elliptic curves ( and number fields) 
Not sure if it will be helpful, but I wrote a survey article whose title could have been "Where do pairings really come from, anyway?" It was for a cryptography conference on pairings. I tried to explain, from a functorial point of view, the origins and relationships of the various pairings on abelian varieties associated with the names of Weil, Tate, Lichtenbaum, Neron, Cassels, ... It's just a survey, so lacks many details, but may be useful in providing an overview. Here's the reference.


J.H. Silverman, A survey of local and global pairings on elliptic curves and abelian varieties, Pairing-Based Cryptography (PAIRING 2010), M. Joye, A. Miyaji, A. Otsuka, eds., *Lecture Notes in Computer Science* **6487**, Springer-Verlag, Berlin, 2010, 377-396.


