Appeal to the head means appeal to people who think with their brains. Rational people. Appeal to the gut means appeal to people who think with their instincts and emotions, not their heads.

Traditionally, the head is the place for thinking, reasoning, being rational, and figuring things out unemotionally. Traditionally, we associate emotions with other places in the body. Love and affection are associated with the heart. Fear and disgust are associated with the "gut," which is an informal name for the digestive system: stomach, intestines, and so on. Intuition and instinctive feelings can be called "gut feelings," too. An example of an "appeal to the head" might be "based on studies, this policy is expected to reduce violent crime by 23%." An "appeal to the gut" might be "Imagine someone breaking into your house and killing your family."