For liability reasons, they release you to the family member or friend who then escorts you.
You are still legally allowed to 'check yourself out' which would be considered 'against medical advice', and god knows what you'd have to sign, but the point is that's not typically the way it works. You are under their care until a responsible party comes in to retrieve you.
This is a common meme, that your insurance might not pay for a procedure if you do something against medical advice. I've had a number of doctors claim it. It's false, thankfully.
A lot of times, you can't. They literally will not discharge you until you can prove you have a viable form of transportation, in the form of a friend or family member who can look after you, or a medically-trained taxi. Until you've set one up and they have released you into their care, they will not let you leave.
I don't think they'll physically stop you from walking out the door, though (that's kidnapping, in fact). But as rconti said, they'll make it very clear to you that you're leaving "against medical advice".
(I suppose they might physically restrain you if they have reason to think that you aren't mentally competent at the moment...)
You are still legally allowed to 'check yourself out' which would be considered 'against medical advice', and god knows what you'd have to sign, but the point is that's not typically the way it works. You are under their care until a responsible party comes in to retrieve you.