Why do mathematicians care so much about zeta functions? Why is it that so many people care so much about zeta functions? Why do people write books and books specifically about the theory of Riemann Zeta functions?
What is its purpose? Is it just to develop small areas of pure mathematics?
 A: People write books about the theory of Riemann Zeta functions because there is sufficiently developed theory and enough applications to warrant a dedicated book, much the same way that people write books specifically about elliptic curves or Schrodinger's equation.
As for the research interest in the Riemann Hypothesis, this MO thread gathers some of its consequences and gives an idea of to which parts of mathematics it can apply. 
And as for more "popular" interest, here's a quote from one of the aforementioned books, Edwards' Riemann's Zeta Function:

The experience of Riemann's successors with the Riemann hypothesis has been the same as Riemann's -- they also consider its truth "very likely" and they also have been unable to prove it. ... the attempt to solve this problem has occupied the best efforts of many of the best mathematicians of the twentieth century. It is now unquestionably the most celebrated problem in mathematics and it continues to attract the attention of the best mathematicians, not only because it has gone unsolved for so long but also because it appears tantalizingly vulnerable and because its solution would probably bring light to new techniques of far-reaching importance.

That's from 1974, and is probably even more applicable today.
A: For one thing, the Riemann Zeta function has many interesting properties.  No one knew of a closed form of $\zeta (2)$ until Euler famously found it, along with all the even positive integers:
$$\zeta(2n) = (-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!}$$
However, to this day, no nice closed form is known for values in the form $\zeta(2n+1)$.  
Another major need of the Zeta function is relating to the Riemann hypothesis.  This conjecture if fairly simple to understand.  It essentially hypothesizes that the nontrivial zeros of the zeta function have a real part of 1/2.  This hypothesis, if proven true, has major implications in number theory and the distribution of primes.
The Riemann zeta function also occurs in many fields and appears occasionally when evaluating different equations, just as many other functions do.  
Lastly, the sum 
$$\sum_{n=1}^{\infty} \frac{1}{n^s}$$
is a very natural one to try and study and evaluate and is especially interesting because of the above-mentioned properties and more.
