This isn't a feminist protest, this an ecologically more responsible (and private) way of disposing of old bras.
In Japan, burnable garbage is placed into translucent plastic bags which in turn are put into neighborhood wire bins for collection so that non-burnables (which are handled separately) erroneously mixed in with burnables can be spotted before creating a problem in the system.
Unfortunately, that means personal items such as undergarments may be visible before the bag is collected. Many women are embarrassed by the thought of their bras being seen by others or even snatched by a pervert, and so cut them up into small pieces - not easy as many of them contain wires.
In the last few years major underwear makers in Japan started programs to recycle such items and they offer opaque bags for women to use with the recycling program. And instead of just being incinerated, the bras are recycled into what industry calls RPF - "refuse paper and plastic fuel".
So, two um.. thumbs up for burning your bras, ladies... after you're done wearing them, of course.
source: Recycling bras kills two birds with one stone - Kyodo News