I'm also interested in why they started it, but I assume that it's simply that they thought BC/AD was too religiously charged. Still, I'd be interested in the details of precisely how it came up.
An initial emphasis was made by Jewish academics in the mid-19th century, for obvious religious reasons. Famously, Raphall used BCE/CE in his Post-Biblical History of the Jews in 1856.
The trend really didn't take hold until the 1980s, however, when it became a matter of political correctness more than anything. Really the switch is happening at the level of institution, not individual, as for instance the change in the official style guide of the American Journal of Archaeology (see 3.12). The academic journals set the style precedent for the authors they publish, essentially forcing them to switch for that particular article. From the journals, the trend spread to institutions like libraries and museums, some of whom also changed over. And from there it spreads to school systems, textbooks, etc.
Personally I prefer BCE/CE -- I think it is cleaner, and easier to understand at a glance for someone who isn't used to dealing with ancient dating. I also like that many publications are trending towards using BCE/CE without the '.' (B.C.E., e.g.), as this is even more streamlined.
There's always room for discussion, but perhaps the section Calendars and timekeeping from our FAQ will answer your inquiry.