My wife had this same question the other day. You're just missing one step: after you tap on the contact, tap the pencil-shaped Edit icon, probably in the top right corner. That will open the contact in editing mode. You can then add a picture.
In Lollipop (Android 5), once you're in editing mode, there's a plus button next to the contact's name to tap on to add a picture. I don't recall what it looked like in earlier versions, but it'll be something similar.