Vanishing dimensions is basically the theory that the universe was earlier in just two dimensions. At some point in the past, these dimensions warped over, and became four. How this happens is yet unclear, but mathematics will have to be really crazy to get that result. http://atramateria.com/the-vanishing-dimensions-of-the-universe/
This is where the above explanation is given.
Yet another theory states that when the universe started out, it had 10 dimensions, and one more 'super' dimension (Edward Witten's hypothesis). However, after the big bang some kind of rift tore the ten dimensional universe into one of four, and one of six (plus one) dimensions. The 4 dimensional universe kept expanding, but the 6 dimensional one collapsed tinier and tinier, effectively 'vanishing'. I'm more convinced with the second, which is also known as the string theory (now, the M-theory), you can find out about it in 'Hyperspace' by Michio Kaku.