The difficult part of this statement is that it can neither be confirmed, nor denied. It sure sounds good, but the inverse - that teams play better in front of their backups due to a higher level of focus - is equally true. How one frames the statement is also important. Whether the claim is that teams play better or that they play more aggressively changes the point.
It may not answer either question, and there are multiple caveats necessary, but here's a look at individual Corsi scores among Flyers defensemen last year in front of Sergei Bobrovsky compared to Brian Boucher. This was accomplished by taking their Corsi with Bobrovsky and subtracting their Corsi with Boucher.
Click the image to enlarge
The sample size is small, score effects have not been accounted for, ice time in front of each goaltender is ignored, as is zone starts and strength of opposition. it probably does not even answer either question. It does, however, show that Andrej Meszaros was nearly 17% better at controlling shots in front of Bobrovsky than he was in front of Boucher while Braydon Coburn was nearly 10% better at controlling shots in front of Boucher than he was in front of Bobrovsky.
I don't see any playing style causes to this phenomenon - Coburn and O`Donnell are stay-at-home while Meszaros and Pronger are big, risk-taking offensive-minded defensemen. I do, however, see the players with the second, third, and fourth highest CorsiRel Quality of Competition playing better in front of Boucher and the fifth and sixth highest playing better in front of Bobrovsky.
I don't think that's the answer, however, but it's all I have. Anybody have any ideas or explanations?