The good reason is to create a standard. A de-facto standard, which anyone can use royalty-free, based on open source software can be a powerful catalyst for a company’s growth. The best example in the recent past is Rackspace, which has now hit hard times but which absolutely took off after it began sponsoring Open Stack, an open source cloud infrastructure, rising from just over $5 per share to a peak of $80 early this year.
The bad reason to sponsor your software as open source is because you need new customers. An example of this is webOS, which HP made open source in 2012 after acquiring it as part of Palm. No one was fooled. The announcement created no lift. HP sold the software, and its development team, to LG of Korea, early this year for an undisclosed (and likely very embarrassing) price.