Research suggests that silent mutations can change phenotypes, how is this possible?

well some specise exhibits codon specific biase where they prefer to use a set of codon instead of the mutated one. This can further lead to affect of splicing or post-translational modifiction.

Although I don't know if this is right, but my inference is this.  The phenotype describes how the organism will look and genotype describes the genetic make-up.  Well although a silent mutation produces the same amino acid, the CODON itself looks different so maybe that is why the phenotype looks different.  

Often a silent mutations make non-silent mutations more likely.  

Lets say you start with a gene with the code CUA which makes Leu.  Any mutation in the 3ed position will code for the same AA, and there is a mutation in the first position which will make the same AA.  However lets say there is a mutation which changes the sequence to UUA.  Now if there is a mutation in the 3ed position, there might be a change in the AA.

Also there are some regulatory that are in the coding sequence and even a silent mutation may throw off the regulation,