What do the checkboxes that appear when editing the card for "Me" do?

It's for selective sharing of details when exporting a contact card, as this Finer Things in Mac post suggests. It allows you to keep the unchecked items more private by omitting them from the sharing aspect of Address Book (soon to be Contacts) on OS X.

Lion’s Address Book allows you to select which details to share with others when exporting your contact card. As long as you have told Address Book which card is yours using the Card > Make This My Card option, editing your card will display a checkbox to the right of each detail you’ve entered. Deselect a checkbox, and that detail will be excluded when you drag your card out of Address Book or click the Share button at the bottom.