To "trace" can mean "to follow a path." Art stores sell "tracing paper," which is thin and translucent. You can see through it. You can put it down on a drawing, and then you use a pencil to trace the line you see, so that you can make an exact copy on the tracing paper. "Trace" sometimes has the sense of "to discover something hidden, by following a trail that leads to it." "He stole money and vanished. The police got his credit card records and were able to trace him. He had moved to Minneapolis, and he was arrested there." "Paleontologists have traced bird fossils as far back as the Mesozoic era." Your passage means "when we try to discover the root causes of lack of self-confidence, we dig deeper and deeper, and find that it originates in mismanaged memory." To "settle down" means "to stop moving." The word "settle" is connected to the words "seat" and "sit." Sometimes it carries the idea of relaxation, perhaps by sitting in a comfortable chair. In your passage it means calming yourself, relaxing, and focussing on your problem.

To trace sopmething is to determine it's origin. "settle down" means as below to stop moving. Here it means to stop doing other things and concentrate on doing one thing.