It's natural that it's simple, since (and I suspect that here I'm about to get skewered) Brainfuck is basically a straightforward implementation of a Turing machine, just as Lisps are basically straightforward implementations of the lambda calculus.

Lest you think I'm completely talking out of my ass, I defer my defense to this random wiki page:

http://esolangs.org/wiki/P%27%27

  > P′′ is a primitive programming language created by 
  > Corrado Böhm in 1964 to describe a family of Turing 
  > machines.
  > [...]
  > Brainfuck (apart from its I/O instructions) is a 
  > simple informal variation of Böhm's P′′. Böhm gives 
  > explicit P′′ programs for each of a set of basic 
  > functions sufficient to compute any partial recursive 
  > function -- and these programs are constructed entirely 
  > from six P′′ words precisely equivalent to the 
  > respective Brainfuck commands +, -, <, >, [, and ].