Anyone can generate any amount of addresses instantly out of nowhere and be paid to them. Lets say you have x btc that you received as payment from some illegal activity. You can make any number of payments to any number of addresses of your choice, some of which are yours (new addresses, so they're not associated with you yet), some of which are not. You still have control over all your btc but nobody knows what is "you" and what is someone else. While it's everything there in the blockchain for anyone to explore, if properly managed the address system does provide a high level of anonymity by current standards.
edit: a btc address is effectively just calculating a hash, behind it you have good old public key cryptography so it's very easy to generate and can even be done offline.
All you have to do is use a mixer.
The advantage that it has over paper money is that I can almost instantly send it anywhere in the world, for cheap.
> According to Wikipedia: User can send his bitcoins on this service, pay small commission, and then receive the same sum of utterly different bitcoins.
If they're paid out to one address it still wouldn't provide the desired level of anonymity, if however they are paid out to multiple addresses belonging to the same user, I would imagine the anonymity level would increase but not totally since one could still add up payments made to different addresses whose total should amount to a given range.