How do people with a split brain experience reality? Some people with severe epileptic seizures have the connection between their two brain halves cut. How appears reality for them, and why does this procedure helps them?
The corpus callosum is the largest of the commissural neural pathways, linking the cerebral cortex of the left and right hemisphere. The sectioning of this tract is called a corpus callosotomy, which disconnects the two sides of the neocortex.

A callosotomy is used in cases of intractable epilepsy, i.e., epilepsy that cannot be treated sufficiently with medication. 

Patients with their corpus callosum severed have been shown to have difficulties in tasks that require lateralized brain areas to work together. For example, the left visual field is processed in the right hemisphere in the brain (and vice versa), speech and language centers are primarily located in the left hemisphere (in most people) and the right hemisphere specializes in visual-spatial processing and facial recognition.

When experiments are done under laboratory conditions, the lateralization of the brain becomes quite evident in these patients. For instance, a compelling experiment is shown in Fig. 1. When a split-brain subject is shown a word in the right field of view, the image is processed in the left hemisphere in the brain. The case shown shows the typical case where the language centers reside in the left hemisphere too. In this case, the subject is able to say what the word was ('face'). However, when the same word is shown on the left, the visual image is processed in the right hemisphere, where the speech centers are less well represented, if at all. In this case the subject cannot verbalize what they saw, but they are able to draw it(!). And in fact, subjects may say they saw nothing at all when a visual stimulus is presented on their left side, but still make a motor response to it (e.g., press a button when instructed to do so when a stimulus pops up).    


Fig. 1. Split brain experiment. source: Nature 

Why does callosotomy help epilepsy patients? When conventional methods fail to treat severe forms of epilepsy (intractable epilepsy), sectioning the corpus callosum may help to prevent the seizure from spreading from one hemisphere to the next. Generalized seizures result in loss of consciousness only when both hemispheres are affected. Often, seizures start in a hot spot, called an epileptic focus. From there it spreads over the cortex and the subject gradually loses consciousness. Once it jumps over, via the corpus callosum, the subject loses consciousness entirely. Hence, severing the major connection between the two hemispheres can keep seizures from evolving into full-blown generalized seizures. However, callosotomies have been really rare in the past and they are hardly performed today, due to the advances made in anti-epileptic drug development, combination therapies, micro surgery, etc.

Resource
- Nature News, March 2012