Our nervous system extends into our bodies. We feel emotions in our bodies. People with certain kinds of brain damage that prevents them from feeling emotions normally also experience trouble making rational decisions.
More recent research has been hinting that we may even hold certain types of memories outside our brains.
Humans have always been drawn to neat, tidy ideas, especially ones that draw clean boundaries: it's an appealing idea that our consciousness lives solely in our brains, and that our brains could function independently of our bodies, but it seems unlikely that it's really that simple.