Hello Olive, In this sentence "sick" means "perverted" or "disgusting". For example, "What? He married his sister? That's sick!!!" Therefore, "a man is only as sick as his secrets" would mean something like this, "if you want to understand how disgusting a man is, see how disgusting his secrets are: that will tell you."

I think it means that you're only as bad a person as the secrets you keep. If you live admitting to something. If you live your life openly and everything is out then it's ok because everyone knows. Basically, it's what people don't say or what they hide from the world that is the most troublesome. You're only as bad or -sick-(if you will) as the things you refuse to tell or won't let anyone know so get it out in the open!