How does CpG island methylation lead to gene silencing?
CG methylation has long been associated with gene silencing due to the generally negative correlation between gene promoter methylation and transcription levels.  

When CG methylation occurs in the promoter or enhancer region of animals (where these 'CpG islands' tend to be), methylation seems to impede (to some extent) transcription factor (TF) binding.  That is the baseline mechanism for gene repression.  The altered chemical properties of the DNA make it less favorable binding site for its cognate TF. 

In some cases, methylation clearly prevents TF binding.  For instance, some TFs bind DNA to block methylation, which facilitates access of another TF or set of TFs to the bound DNA (for instance, this paper by the Schubeler lab about NRF1).  In the case of other TFs, their binding is less reliant on methylation state, possibly due to the strength and breadth of the binding site (for instance this work from the John Stam. lab about CTCF).

Methylation also can affect nucleosome positioning and stability in some contexts; the nucleosome effects are poorly understood currently  because nucleosomes and methylation likely feedback on one another.  See this work from the Zilberman lab for an example.

The role of a higher-order repressive structure forming is arguably not obvious yet; for instance, histone H1 bining does not appear to be heavily influenced by methylation state (although this is somewhat contentious because there is often a positive but weak correlation at repressed elements in the genome between H1 and methylation--for instance just pubmed 'h1 chip' with reviews and read a recent one).  More certain is the relative change in location of methylated promoters within the nucleus to a place of transcriptional 'silence' (inactivity).  These repositioning phenomena are under intense investigation; you can learn more about this by googling for 'nuclear topology and gene repression' or something like that.  

hope this helps