1) The underlying companies of the BSA (incl. AAPL and MSFT) were not really in favor of SOPA to begin with, and the public attention has lead them to correct the initial BSA position.
2) The underlying companies of the BSA were in favor of SOPA, which is why BSA supported SOPA. But in the face of public backlash they are backpedaling, because it is not valuable to support a bill that will lose regardless.
Comments
Let's distinguish two possible theories:
1) The underlying companies of the BSA (incl. AAPL and MSFT) were not really in favor of SOPA to begin with, and the public attention has lead them to correct the initial BSA position.
2) The underlying companies of the BSA were in favor of SOPA, which is why BSA supported SOPA. But in the face of public backlash they are backpedaling, because it is not valuable to support a bill that will lose regardless.
The second seems much more likely.