What types of proof are there of this result and where can I read about it?
I think that the Hardy-Littlewood circle method can prove that every number is the sum of something like $100000$ cubes, and you can use tables to prove those "small" numbers are all expressible as sums of cubes.. which gives you warings problem but I was more interested in specific proof about the cubes.
Dickson showed that the only integers requiring nine cubes are $23$ and $239$. Wieferich proved that only $15$ integers require eight cubes: $15, 22, 50, 114, 167, 175, 186, 212, 231, 238, 303, 364, 420, 428,$ and $454$ from mathworld
Best Answer
There is a paper by L E Dickson, Simpler proofs of Waring's Theorem on cubes, with various generalizations, from the Transactions of the American Mathematical Society for 1928, available here. But you may be disappointed --- "Simpler" doesn't mean "simple".