The phrase is often used after someone experiences something unpleasant that causes them to focus inward. He didn't talk much at the party; it's like he turned in on himself after the argument.
Pro Tip 2/3
Isolation Indicator
The phrase suggests that a person may start avoiding social interactions. Ever since she failed her exams, she has turned in on herself, not even joining us for lunch.
Pro Tip 3/3
Emotional Undertone
It implies a deep emotional process, not just ordinary self-reflection. Losing his job made him turn in on himself, questioning his worth.