At the 11:00 minute mark of Season 1 Episode 7 Stick, the scene is a flashback to when Stick and Matt Murdock first met.
It's at an orphanage run by nuns. It starts out with Matt writhing in pain on his bed. Then, it cuts to Stick walking alongside a nun. They speak:
Nun: He's getting worse. When we first took him in, Matthew's problems seemed less severe. Now he's in so much pain. Doctor's, clergy, no one has any idea what's wrong with him. We heard about your work with special children and thought maybe you could help.
Stick: Not for free.
So, it seems that Stick has a reputation for helping "special" children. Likely, this is due to his involvement with the Chaste and trying to find new recruits with special abilities.
Stick himself later hints at this being part of what he does, in a conversation with Matt. Again, this happens during a flashback, with the relevant dialogue starting around the 16:35 mark of the same episode:
Matt: If I ask you a question, will you hit me?
Stick: Depends on the question.
Matt: How did you find me? How did you know?
Stick: Old bitty at the orphanage thinks it's her idea.
Matt: But it wasn't, was it?
Stick: Maybe it's one of my gifts. Or you just got lucky.
Matt: That's not an answer.
However, here, the implication is that the nun reaching out wasn't actually a factor in Stick finding Matt. This further pushes the idea that there is some mysticism or work of The Chaste going on here.
But, as young Matt so clearly told us, we don't actually get an answer.
This, obviously, only accounts for the Marvel Cinematic Universe. I can't provide an answer for the comic continuity.