...I recommend it - because it hides your name, address, phone, and email address from people who would send you SPAM related to your domain.
I recently registered a new domain (abcinc.biz) and forgot to add privacy protection. I began getting 2-3 offers a day offering to build my website. (Most offers were from India, BTW)
I have since added privacy protection, and I hope the damage will be contained. I'll be curious to see if these people sell/resell/share/trade my address with friends and competitors. Ugh.
You provide information when you create an ABCInc customer account. We don't use it for anything other than to contact you and to fill out the WHOIS information to register your domains.
The WHOIS registry is open to the public, and that's where your information escapes into the Internet.
When you purchase privacy protection with us for your domain, we substitute a 3rd party intermediary for the WHOIS registration, and filter out the garbage.
[Update: 4 months later I'm still getting SPAM, and it's intended to be scary.]
Here are sources for more information:
(All of the links listed below will open in a new window.)
Privacy Protect.org provides the most concise explanation:
This site examines the arguments both for and against signing up for privacy
GoDaddy charges $7.99/yr for the service (Jan 2016):
You can follow the search I made on Google to see a lot of different explanations.