Intersectionality (and yes, the name's awful) was an improvement on the more blanket "oppressor/oppressed" views that preceded it or emphasized one aspect (class or race, for example) to the exclusion of others. I think Leigh's got some good points about it checking some pretty rampant assumptions.
Voices getting ignored can seem abstract (and the academic jargon surrounding this theory doesn't help) until cops won't charge a perp because their culture says he's more sympathetic than the person who was actually harmed. Seeing cases like that, over and over, made me believe intersectionality is quite real.
James is right: it's a tool, and the bullshit part comes when it's used badly. At its best, I've seen activists use intersectionality to analyze where they need to hit, to better understand how a mix of culture, policy and power make a specific institution horrific on a particular issue they want to change. That's not individuated and fragile, but a way to organize more effectively while not dismissing the complexity of what's actually going on or leaving out a whole affected group. Personally, I've found intersectionality handy in better understanding complicated topics like rape culture at universities.
Unfortunately, protest culture largely hates the level of practical strategy that makes intersectionality really useful, so many times when the idea gets brought up in day-to-day politics, it's bogged down in endless hair-splitting. One of the smartest activists I've ever known said that the most effective saboteur wouldn't be the provocateurs protesters get paranoid over, but someone who says "privilege" in response to every matter, regardless of relevance. She was completely right.
While I oppose restrictions like parade commissions and hate speech laws for obvious free speech reasons, those rights don't protect people from a public backlash: it's perfectly legit to try to stigmatize your opponents to the point where they lose practical influence.
Some radfems remain surprisingly influential in pushing really awful policies, and have helped make life hell for trans people for a long time. Their opponents are well within their rights to bring down bad publicity when they meet.