How did genome duplication in jawed vertebrates allow gene specialization?
With a single copy gene, mutations may affect some vital function and are therefore often deleterious and negatively selected. Immediately after a duplication event, there is a level of redundancy in gene function and mutations in one copy giving rise to novel phenotypes are more easily tolerated since the ancestral function is retained by the paralogous gene.

Some reading material, if you feel so inclined:



Wikipedia:

Functional Divergence


  Functional divergence is the process by which genes, after gene duplication, shift in function from an ancestral function. Functional divergence can result in either subfunctionalization, where a paralog specializes one of several ancestral functions, or neofunctionalization, where a totally new functional capability evolves.


Neofunctionalization


  Following the gene duplication event there are two identical copies of the ancestral gene performing exactly the same function. This redundancy allows one the copies to take on a new function. 


Subfunctionalization


  Specialization is a unique model of subfunctionalization, in which paralogs divide into various areas of specialty rather than function. In this model both gene copies perform exactly the same ancestral function. For instance, while the ancestral gene may have performed its function in all tissues, developmental stage, and environmental conditions, the paralogous genes become specialists, dividing themselves among different tissues, developmental stages, and environmental conditions.




Kryuchkova-Mostacci N, Robinson-Rechavi M. 2016. Tissue-Specificity of Gene Expression Diverges Slowly between Orthologs, and Rapidly between Paralogs. PLoS Comput Biol 12:e1005274.


  The most widely accepted model is that orthologs diverge slower, and that the generation of paralogs through duplication leads to strong divergence and even change of function.




Soria PS, McGary KL, Rokas A. 2014. Functional divergence for every paralog. Mol Biol Evol 31:984-992.


  Because genes can be constrained by selection at more than one phenotypic level, the relaxation of constraints following gene duplication allows for functional divergence (FD) along multiple phenotypic axes.