Why are mathematical results discovered by multiple people independently?
The same thing happens in science generally. The science historian Thomas Kuhn wrote a famous essay about this phenomenon, "Energy conservation as an example of simultaneous discovery", in The Essential Tension; you may want to take a look at it.
As long as we believe that mathematics exists in some sense independently of people, I think it's not so surprising. Take the discovery of calculus. The basic problems of calculus (finding a tangent, finding the speed of a moving object, finding areas) had been around for a long time. In some form, the ancient Greeks worked on these problems. In the generation before Leibniz and Newton, algebra reached pretty much its modern form, at the hands of Fermat, Descartes, and some others. To a very large extent, calculus is what you get when you mix together the classic problems with the symbolic techniques of algebra, and stir vigorously.
As another example, look at the constructions of the real numbers: Cantor and Dedekind. Mathematicians like Euler, the Bernoullis, Lagrange, and Laplace took the calculus and developed it extensively. Inevitably, the logical problems and fuzzy spots came to the surface. Already with Gauss, Cauchy, Abel, and others you can find complaints about the lack of rigor. So there was a perceived need for a more precise definition of what the real numbers "really were". On the one hand, it's not surprising that the previous generations hadn't worried too much about this: they were having too much fun exploiting the legacy of Newton and Leibniz, and the problems hadn't become acute. A perceived need, and a couple of geniuses: voila, a solution.
Note however that Dedekind and Cantor gave different constructions. For that matter, Newton's calculus differed in many ways from Leibniz's. This is generally true of simultanous discovery, when it's examined more closely. Kuhn discusses this in detail.
Why are mathematical results discovered by multiple people independently ?
Why is the sun discovered by multiple people independently ? Why do two people who look in the same direction see the same thing ? What happens when you take the Taylor series formula for the exponential function, and switch the base and the exponent in the numerator ? You rediscover the Bell numbers, whose roots date back to medieval Japan. What happens when you try to introduce a symbolic notation for nested radicals, similar to $\displaystyle\sum$ and $\displaystyle\prod$ , for instance, and then you write a negative quantity for its order ? You rediscover the fact that nested fractions are nested radicals of order $-1$, about a century after Herschfeld. What happens when you take the binomial theorem, and place a non-natural quantity for its exponent ? You rediscover the binomial series, centuries after Newton. What happens when you play around with definite integrals whose integrand does not possess an elementary anti-derivative, and you start focusing your attention on $\displaystyle\int_0^\infty e^{-x^n}dx$ ? You rediscover the expression for the $\Gamma$ function centuries after Euler and Gauss, by zooming in on its behaviour for $n=\dfrac1N\in(0,1)$. Etc. And the list could go on $($and on, and on$)$. It's all just one giant inter-connected web of lies, uhm, I mean, truth. ;-)