"Dues" in this phrase are not "deudas" but "derechos", as in "derechos de entrada", if there is such an expression.
"I've paid my dues" is generally used figuratively to mean that I've done the work, made the effort, spent the time necessary to earn whatever status or position I have achieved.
Does that help?
Tom: Ok, I finally finished helping you move all the boxes
Elizabeth: Well, I think you should help me move the furniture too, after all you did tell my big secret.
Tom: I don't think so, I've paid my dues.
"I've paid my dues" means "to be even with someone" or like BrilliantPink said, do the work, make the effort, or spend the time. For example if someone does something for me and I do something for them in return, we are even, and I no longer owe him anything because "I've paid my dues".
Paying your dues is when you earn something,like a position in a job.If someone comes along and gets the job, but hasn't earned it,say they're given the position because they're the boss's cousin or something like that.They didn't earn (pay their dues) for the position.Hope that helps ,lockpicker