Apart from telling that the hypergeometric functions (or series) are the solutions to the (essentially unique?) fuchsian equation on the Riemann sphere with 3 "regular singular points", the wikipedia article doesn't illuminate much about why this kind of special functions should form such a natural topic in mathematics (and in fact have been throughout 19th century). Simply:

What are hypergeometric series really, and why they should be (or have been in the past centuries) important/interesting?