When you would normally sign a printed receipt, you sign with your finger on the iPad with the Square app (most implementations I've seen have iPads on a swivel stand that turns around to face the customer). In the example video they show a customer being charged via Card Case which is the more integrated form of transaction but not the common case.
I've seen this a handled a few ways, depending on the merchant.
1. They show me the price before submitting it. I ask for a receipt sent to my email, which they need to type in, but I get it on my own phone immediately. The clerks I've seen do this HATE having to type in my email address because they invariably type it incorrectly.
2. The ipad swivel. I sign with a stylus, type in my own email address and choose to send the receipt that way. The whole time I'm talked through it all by the merchant. I prefer this method.
3. The clerk just rings me up and never bothers to offer the receipt, but I ask anyway, because, like the grandparent post, I don't quite trust these things for some reason yet. The clerk has no idea how to do it, so I end up showing her and then tell her about the swivel mounts and the stylus.
From my experience of paying with Square, I only had to manually enter my email + phone number the first time I paid. Every time since then it's been automatically populated. Swipe, sign and the receipt is in my email second later.
