I am by far not an expert, but I read the paper and maybe I can help clarifing things a bit.
Let's start with the simplest answer to your question:
In the last paragraph of the Introduction paper the authors say
"Throughout this manuscript [...]'background networks' and principle components are used interchangeably." ( Torkamani and Schork, Genetic background and drug response, The Pharmacogenomics Journal (2012) 12, 446-452; doi:10.1038/tpj.2011.35)
So they did not "use" PCA to get the background networks, they just call the PCs background models to indicate how they interpret it: As stated earlier in the introduction
"...interacting networks cannot be expected to correlate strongly with drug response, as their influence may only be observed when the major determinant of drug response and the interacting network complement one another or are both at a synergistic state. A major problem with identifying these interaction partners [...] is the extremely large number of possible partners [...] and [...] that individual genes are unlikely to accurately represent the overall state of a biological network." 
As far as I understand the article, they suggest PCA as a kind of compromise: Ignoring interactions would miss associations of all networks lacking a single gene representing the network's state accurate enough. Including all interactions is infeasable due to the huge number of gene-pairs. By PCA, the number of interactions can be decreased by orders of magnitude while keeping a maximum on information (as in variation): Instead of using the interaction (as in product) of all probes with all probes, they only consider those of all probes with the first six PCs.
I think, in this context 'background' is not used in the meaning of 'background noise' but of 'cultural background' - instead of assesing individual interactions, genes are assigned a 'background'. Since this corresponds to a) the biological network(s) they belong to and b) the principle components(s) they contribute to, those two ideas are used synonymously by calling PCs 'background networks'.