I think the "old souls" are the ones who have experienced more emotional pain and disappointment all throughout their lives than their peers could ever imagine. During experiences of pain and suffering, you're required to examine and discover what's inside the deepest depths of yourself, and become stronger just to survive. In turn, it increases your passion for life and makes you appreciate everything, big or small, so much more.
When you come across people in life who've been through similar experiences, other "old souls", you don't have to ask, you just "know" because it pours out of who they are.
Here's a quote I found that says what I was trying to say:
“When we don’t turn away from pain, we open our hearts and are more able to connect to the best part of ourselves and others–because every human being knows pain. I’m not sure what enlightenment is, but I’m sure it has something to do with turning pain into love.”