FastICA - Purpose of centering and whitening the Data
The goal of FastICA is to rotate your data (unitary transform) so that each axis looks as non-Gaussian as possible.  Gaussian data still looks Gaussian when you rotate it.  

If you don't "sphere" the data, all the algorithm can really do is rotate the whole block to one axis.  By bringing the mean to zero (centering), and normalizing the variance in all directions (whitening), you give the algorithm freedom to rotate in all directions.  

This lets FastICA find the rotations that correspond to non-gaussian data.  All of the rotations it tries will leave the Gaussian components of your data as zero-mean and unit-variance, but non-gaussian components can be optimized.