The phrase signifies offering emotional comfort to someone who is upset. After he lost his job, he just needed a shoulder to cry on.
Pro Tip 2/3
Metaphorical Use
The phrase is metaphorical; you don't actually cry on someone's shoulder. Whenever I'm down, my dog becomes a shoulder to cry on, metaphorically speaking.
Pro Tip 3/3
Informal Expression
It's an informal expression, used more in personal conversations than formal settings. Feel free to call me if you're ever in need of a shoulder to cry on.