How or why is Universal Grammar gaining importance in the field of Linguistics?
Sophisticated brain research, like fMRI and the even more sophisticated positron emission tomography, have shown that early theories of distinctly hierarchical divisions of brain function areas, e.g., Broca and Wernicke areas (related to production and understanding of speech) are incorrect. Research has shown that, in contrast, brain activity is far more diffuse and far less schematized than early technologies showed.
Universal grammar attempts to draw grammar away from schematized syntactical analysis of what is to a biologically based description of brain function within the cascade of diffuse operations now being mapped out. In other words, universal grammar will tell not what we do with language but how language comes to be by identifying and defining the processes that are common to all languages.
Universal grammar is gaining importance through (how) the rapid technological advances that make finding a unified theory of language structure plausible. It is gaining importance because (why) of what decoding universal grammar can contribute to understanding the organic biological nature of cognitive thought. The ultimate application is expected to be an understanding of all biological information transfer within synaptic pathways and within genotypes.
[For more detail, see "Universal Grammar" by Charles Henry of Rice University, Houston, Texas, published in Communication and Cognition - Artificial Intelligence, Vol. 12, Nos. 1-2, pp. 45-61, from which this answer was drawn.]