Confusion related to a term probe-by-background interaction
The "probe by background" interaction is the response of different probes as a function of background gene expression. For example, depending on which of the 6 backgrounds a probe is in, the drug response may go up or down. Probes as a function of background is probably easier to imagine than background as a function of probe (which is equally valid). For 39,115 probes and 6 background networks, there are 234,690 interactions.

In a technical sense, "probe by background" is the interaction term in the linear model that the authors fit for each drug. The details of the analysis are in the supplemental methods. The model they fit is

Drug_Response ~ Probe + Probe * Background + Background


which translates to "Drug response is modeled by Probe, the interaction of Probe and Background, and Background". The middle term (with the *) is the interaction term. Here is a page that helps to explain how to understand significant interaction terms in linear models.