Biological meaning/interpretation of clustering coefficience
First of all, I find this an interesting question.

In short:


Clustering coefficience is widely used
It is widely accepted that hub genes are more essential than poorly connected genes
but research is not always interested in the hub genes / proteins / molecules (for example most metabolic pathways involve water)


Answering your question, why people would want to use this kind of method:
If you would take a random model (pathway) most of the lines will be equally distributed between the nodes. This is not the case for most biological networks. Because of this, the global clustering coeffiecence can be used to see if a network could be biological. Clustering coeffiecence can also be used to find out about more specific nodes. For example if a protein has a relationship with two other proteins (binding, regulation, etc.), the two other proteins are also more likely to have a relationship with each other (similar to a social network).

References:

http://en.wikipedia.org/wiki/Clustering_coefficient

http://www.biomedcentral.com/1752-0509/6/34

http://www.pnas.org/content/99/12/7821.full.pdf+html