On Siegel mass formula

Try this reference https://arxiv.org/abs/1105.5759 (Notes on "Quadratic Forms and Automorphic Forms" from the 2009 Arizona Winter School, by J. Hanke).

P.S. See also http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.543.8732 (A proof of Siegel’s weight formula, by A. Eskin , Z. Rudnick and P. Sarnak) and https://www.youtube.com/watch?v=b3qDTu0C7dM (a video of Jacob Lurie's lecture "The Siegel Mass Formula, Tamagawa Numbers, and Nonabelian Poincaré Duality").

One more interesting paper is https://www.researchgate.net/publication/2404971_Low-Dimensional_Lattices_IV_The_Mass_Formula (Low-Dimensional Lattices IV: The Mass Formula, by J.H. Conway and N.J.A. Sloane).

There are many, many references on quadratic forms. This is a huge area, depending on which way you want to go. One of the main approaches is to construct a theta series associated to your quadratic form whose Fourier coefficients give you the representation numbers you want. These are modular forms, and this is treated in many introductions to modular forms (in fact, I have some notes on this).

One specific reference which I think is nice is the book:

Topics in Classical Automorphic Forms, by Henryk Iwaniec.

He discusses the Siegel mass formula in a simple setting and explains how you can use can get formulas for representation numbers in special cases. In general, you can get "explicit formulas" for representation numbers in terms of Fourier coefficients of Hecke eigenforms. You can get nice asymptotics on representation numbers this way, though the precise arithmetic of Fourier coefficients of cuspidal eigenforms is mysterious.