Visceral literally means of or related to the viscera, which dem bones has admirably defined here as "Innards, the guts. Bowels, intenstines...instinct" (it is also the plural of viscus, flesh). So, Glowing Fish's contention that it means the interior of the body is not exactly correct; it refers more particularly to the internal organs, once thought to be the seat of the emotions.
Building on this understanding, visceral is used figuratively to mean inner or instinctual. Visceral refers to emotions or ideas that are very deeply and intimately seated, intuitive and unreasoned, even crude and elemental. Visceral is thus the opposite of intellectual or elevated - by reason or otherwise. It refers to the earthy and the deeply felt at a gut level.