Bryan Birch's view is that they form a bottomless area for research problems. All answers to the question fall into two types: showing examples of why this is true, and asking why it should be true. G. H. Hardy by temperament seems to have been convinced of the first part by the tau-function, while banning reference to "modular forms" in the general theory of Hecke, which was the 1930s answer to the second part. This all came round again in the 1960s, with the BS-D conjecture on the one hand and Langlands on the other. Most answers to why "special functions" are special are question-begging, because intrinsic interest in mathematics can't really be faked, and "symmetry" as an answer doesn't have a definitive formulation. It looks like the Shimura variety concept will have a big explanatory value in future mathematics, but we can't anticipate the real answers.