As I understand it, the notion that the left brain has to do with language is from a rather interesting phenomena that occurs in individuals where there are few or no neural pathways between the two sides of the brain.
Except for Siamese cats and albinos, the left side of the brain processes the right field of view and the right side of the brain processes the left field of view. The way this works is that the bundles of axons from ganglia cells in the eyes split in the optic chiasm. The axons from the ganglia cells in the left side of each eye continue back to the left visual cortex and the axons from the ganglia cells in the right side of each eye continue back to the right visual cortex.
Thus, the left visual cortex processes the right field of view and the right visual cortex processes the left field of view.
From the visual cortex of each side, neurons pass the information to different parts of the brain. In particular, there is an area called the left fusiform gyrus which is believed to associate objects with the names for objects. (There appears to be some other areas that might arguably be involved as well or instead of the left fusiform gyrus.)
In some individuals, there are few or no nerves carrying the signals across between the halves of the brain. In those cases, the only information the left fusiform gyrus receives is from the left visual cortex which processes the right field of view.
Thus, someone with this problem may readily be able to name objects in their right field of view, but not in their left field of view. If they move to another spot and look back where the objects that were then in their left field of view are now in their right field of view, and vice versa, they can then associate names with objects that they were unable to identify before.
As I understand it, this is one major reason for the notion that the left brain handles language.
Interestingly enough, the right fusiform gyrus is thought to identify faces with names. If you have problems with it, then you have prosopagnosia or face blindness. I assume that if you have a split brain as described above, you would not be able to associate names with people in your left field of view without other cues such as the sound of their voices, but you would be able to identify the same people in your right field of view.
The brain is adaptive enough that in the case of Siamese cats and albinos where the retinal ganglia do not cross in the optic chiasm, the visual cortex of each side processes the entire field of view but with only half as much information from each side. Consequently, their left and right fusiform gyri receive data about the entire field of view, not just one side or the other.
As I said, this is supposedly what gives rise to the notion that the left side of the brain is responsible for language. But in reality, if you do not have the connections between the halves of the brain, your so-called "left brain" or "right brain" activities will suffer because it really takes both side of the brain, not one.