It's COPPA. https://en.wikipedia.org/wiki/Children%27s_Online_Privacy_Pr...
Some people even consider COPPA to be a form of censorship. It creates such a huge legal burden on websites if they allow < 13 year olds, that most websites have to ban them, thus limiting their right to speech.
Obviously it depends on the jurisdiction, but typically minors cannot enter legally binding contracts. Some form of source:
http://www.lawchek.com/Library1/_books/contract/qanda/legala...
One interesting feature I wasn't aware of is that this does not generally apply to "necessities" or employment contracts.
I thought all the age-of-13 stuff, specifically, was because you can't legally enter yourself into a contract when you're younger than 13. Which is to say, you can't accept EULAs or Terms of Service (or you can, but they'll have no legal force.)