What are the stories Rick heard in Casablanca?

Having just found GradeSaver, a cool resource for quote analysis, the entire quote is

I've heard a lot of stories in my time. They went along with the sound of a tinny piano in the parlor downstairs. 'Mister, I met a man once when I was only a kid,' they'd always begin.

The line was classified as a

"quite a definite reference to a bawdy house", or a brothel, which was, at the time, prohibited [in LA].

However, the studio refused to remove it. It's interpretation is that

[Rick] blames her for his pain, and this line describes the depths that he fell to after she broke his heart - looking for female companionship in brothels while Ilsa was standing by Laszlo's side.

So these are stories told by prostitutes about how they became sex-trafficked children, and serves to show the audience how low Rick descended, and how angry he was (and still is!) at Ilsa.

The line "Mister, I met a man once when I was only a kid" is not a reference to brothels or child sex trafficking. In Casablanca, Rick frequently calls Ilsa a "kid". "Here's looking at you, kid." Etc. In the 1940s and 1950s in was very common in movies for young adults, both male and female, to be referred to as "kids" by older adults. One example: Danny Kaye in White Christmas refers to young adults as kids more than once, and they're obviously not children. If I could be permitted to rewrite Rick's statement into modern American 2019 boring parlance, it would go something like this:

"'Mister, I met a man once when I was just a young woman,' they'd always begin."

The tone and implication is that young people often fall in love with individuals who, later in life, they would look back on as unsuitable, or foolishly chosen, or that the romance was a mistake. That's all this quote means. Respectfully, BlueMoon93's answer is way off.