"Anonymizing software can provide a key layer of protection by routing your communication with most web sites through an anonymous proxy server that substitutes its Internet address for yours. That way, you never actually "touch" the web site; the proxy does it for you. And so your private information is kept private. "
So now instead of servers you communicate with each having a bit of information about you a single place on the web run by someone who may or may not be trustworthy has all of it.
You've just sent your SSN, creditcard number, mother's maiden name, all your passwords, address and telephone numbers, the application letter for that job requiring a security clearance, and your tax paperwork all to someone who might well decide that they can make a fair sum selling it to someone, or they might use it for some nifty identity theft themselves.
That's the dark side of these "anonymisers". They prey on fear of some dark conspiracy to get all your information to do just that, get all your information.