Your gut feeling (or instinct) is driven by your emotions, not your logic.
Logic is good for simple systems. Instinct is good for complex systems where logic fails to understand everything.
Sometimes you will make a decision based on your gut feeling that goes against logic and it will lead to the best decision of your life. Like moving abroad when everyone around you is telling you that you should stay for this or that reason.
Patterns in complex systems are obvious in retrospect. Same is true for gut feelings. Your decision is not obvious when you take it, but it becomes obvious why you took it only later on.
It's hard to convince someone of a gut feeling