A person is only as sick as his secrets is referring to the person's morals and values, in a sense. The "sickness" isn't physical; instead it is referencing his state of mind.
For example, if you were to do something that you were ashamed of, the guilt would make you feel badly about what you had done. Or, if you didn't feel any guilt, it could be that you would attempt to have other people do those things that you do. The sickness is the behavior, in this instance.
I didn't see the show you are referencing so I am not exactly sure what the context of the statement entailed.