Yes, the result is very mathematical, though computers were used for many of the calculations. Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge proved this result in July 2010 and put up a very nice webpage at cube20.org, with some history of the problem and their basic methods.
The team consisted of a programmer, a math teacher, a mathematician, and an engineer. There are two parts to the problem: find a method that lets you solve any cube in M moves or less, and find a really hard starting position, that takes at least N moves to solve. These are upper and lower bounds for the problem-- the goal is to keep finding better methods for solving cubes, or to keep finding tougher starting positions, until you can show M=N.
In 1981, it was known that some positions took at least 18 moves, and Morwen Thistlethwaite had a method to solve any cube in 52 moves or less. The website lists the progress over the last few decades, narrowing in on the number 20. Part of the progress comes from having more powerful computers to work with, but a lot of it is having a better idea (math!) to simplify the problem.
Here's a cool table (copied straight from cube20.org) showing how many starting positions require 20 moves, 19 moves, 18 moves, etc. You can see there are many, many starting positions; and it's actually really tough to find one that takes 20 moves to solve.
Distance Count of Positions
16 about 1,100,000,000,000,000,000
17 about 12,000,000,000,000,000,000
18 about 29,000,000,000,000,000,000
19 about 1,500,000,000,000,000,000
20 about 300,000,000
As for bigger cubes, there are many methods for solving bigger cubes already. It's actually not that much tougher than solving the 3x3x3 cube; it just takes longer. But I don't think many people will work on finding the optimal number of moves for bigger cubes-- it will be much harder (probably not possible with current computers) and is not as interesting (because everyone cares most about the 3x3x3 cube).