Drafts seem to save if a message isn't sent. I thought it was odd too until I figured it out by accident. Just type a message but don't send it, then go back to it later. The other thing is it has to be an existing conversation, you cannot start a new conversation to someone saved as a draft, Apple will not allow that.
Google Messages shows you there is a draft saved when you back out of the conversation, really nice elegant touch too, the draft message is in italicized font.
iMessage leaves so much to be desired, it really speaks about the users who have tolerated it for so long. For example that green background with white text when it's an sms, how can people put up with that combination for so long? I guess you really don't know what you're missing when you don't know other options exist.