Understanding sphere packing in higher dimensions
There are two things you need to understand. The first is how to prove sphere packing bounds via harmonic analysis ("linear programming bounds"). My lecture notes from PCMI 2014 give an exposition of this theory, which covers the period up to, but not including, Viazovska's paper on eight dimensions. This reduces the problem to finding radial functions in eight and twenty-four dimensions with certain roots and sign conditions, and the lecture notes discuss why everybody believed such functions exist. The functions themselves are surprisingly subtle; see this paper with Steve Miller for numerical experiments and conjectures.
Then Viazovska's breakthrough lies in how to construct the right functions. It's convenient to split them up into eigenfunctions of the Fourier transform, which have to have roots at certain locations. One way to look at it is that she brute forces those roots by including a sine squared factor that produces the roots, and then she multiplies it by something that's essentially the Laplace transform of a (quasi)modular form. To make this work, one has to answer three questions:
What sort of modular forms play nice with the sine squared factor and produce radial eigenfunctions of the Fourier transform?
Do there exist such modular forms that actually give the desired functions?
How can we prove the inequalities these functions are supposed to satisfy?
Viazovska's methodology gives nice answers:
This amounts to identifying the right level, weight, and depth (for quasimodular forms).
Yes. One can reverse engineer them based on the desired properties.
In the best way one could hope for, namely at the level of the modular form itself (rather than having to worry about something subtle happening in the Laplace transform).
For details, see the eight and twenty-four dimensional papers. As Peter Sarnak points out in Erica Klarreich's Quanta article, the actual construction is surprisingly simple. If you know about linear programming bounds and basic facts about modular forms, the proof itself is very down to earth.