How people comprehend ‘natural language’ like stories, texts, and conversations, Is a complex behavior, calling upon numerous cognitive and neural systems. A great deal of research has focused on how people decode written and spoken input into words, retrieve word meanings, and parse syntax.

For over a century, the consensus has been that frontal and temporoparietal regions (where the temporal and parietal lobes meet) of the Left Hemisphere (LH) are crucial for these fundamental language processes.

Recent investigating on how the brain converts a continuous stream of words into meaningful communication has proceeded more slowly, but is beginning to reveal how the brain comprehends natural language. One conclusion is that, as language input gets more complex, there is more involvement of Right Hemisphere (RH) homologues to classic LH language areas.

What is the real difference between Left & Right hemisphere?

The main difference between LH and RH can be found at the micro-anatomical level. There are numerous asymmetries that support distinct computations within a process, conferring distinct advantages to each hemisphere:

• The LH strongly activates small and focused semantic fields, containing information closely related to the dominant meaning of input word(s).
• The RH weakly activates large diffuse semantic fields, including information distantly related to the words, providing only a coarse interpretation, insufficient for many language tasks.

• Pyramidal neurons in RH language areas have longer initial dendritic branches and more synapses further from the soma. This causes RH neurons to receive a broader and more overlapping field of inputs. 
• The LH cortical columns are more widely spaced, with less overlap among input fields.
• The larger semantic fields of the RH are more likely to overlap, allowing weak activation to summate, when input includes multiple distantly related words. Therefore, the RH is sensitive to distant semantic relations, and comprehenders capitalize on this sensitivity when understanding natural language, particularly figurative language or unusual constructions.

These asymmetries are also consistent in cortical micro-circuitry of language areas that influence how neurons spread information. For example, at the cellular level, pyramidal cell dendrites branch further from the soma and ultimately into more branches with more dendritic spines, on average, in the RH than in the LH. Such circuitry favors more input from relatively distant sources in the RH, and from close sources in the LH.

This is a clip of how Neurons, branches and networks work:

Because functional and structural levels of brain organization are interdependent, some effects at each level get passed through to higher levels. Thus, cortical mini-columns, macro-columns, and functional areas are more highly overlapping and more densely interconnected in the RH than in the LH. Overall, these microcircuitry asymmetries suggest broader input and projection fields, and greater functional overlap across processing units in the RH than in the LH. The images left are photomicrographs of microcolumnar and macrocolumnar structures in the human temporal cortex, where you can notice the columnar arrangement of pyramidal cells in the upper cortical layers. Bundles of afferent and efferent fibers traversing vertically between the columns of pyramidal neurons.

Sources:

• Bilateral brain processes for comprehending natural language

  by Professor Mark Jung Beeman, Ph.D source-pdf 

• Hemispheric asymmetries in cerebral cortical networks

  by Professors Jeffrey Hutsler and Ralf A.W. Galuske  source-pdf