Is this slang?
To gut means to remove the inside organs (heart, stomach, liver). Deer are gutted when processed. The "guts" are all the insides.
So when a person says "I am gutted" it means that it may feel like their insides are being torn out. A very deep, emotional feeling.
In fact, the term "gut wrenching" is a term used to describe a deeply profound event or feeling. To see children killed in war is gut wrenching.