I can't speak for applications in physics, but in combinatorics and analytic number theory there is no magic involved. The idea in combinatorics is that you start by looking at the ring of infinite sequences over C with componentwise addition and Dirichlet multiplication as the product. That is, if a, b : nat -> C we define

(a <*> b)(n) = \sum_{k | n} a(k) * b(n/k)

It is easy to check that this is a ring, and it has wonderful properties which make it very easy to solve many equations of interest in this ring. This is usually called the ring of "arithmetic functions".

However, it is difficult to get asymptotic estimates for the coefficients of a series by purely algebraic means. This is the first and last time that complex valued functions enter the picture, but it's a very neat trick. Let's consider two series a, b and define the functions

A(s) = \sum_{n >= 1} a(n) * n^-s
B(s) = \sum_{n >= 1} b(n) * n^-s

then we have

A(s) * B(s) = \sum_{n >= 1} (\sum_{k | n} a(k) * b(n/k)) * n^-s = \sum_{n >= 1} (a <*> b)(n) n^-s

So this mapping, from the ring of arithmetic functions to the ring of (partial) complex functions with pointwise addition and multiplication. Glossing over some details for now, this allows you to analyze the function belonging to a sequence to gain information about the sequence itself. In particular, you can use the theory of complex integration and Cauchy's residue theorem to gain information about (partial sums of) coefficients.

Unfortunately, the world is not quite this simple. The functions we are mapping into typically aren't very well behaved and usually aren't defined on large parts of the complex plane (consider a(n) = n^n). This means that all of our nice tools from complex analysis actually won't work very well!

The whole idea behind "analytic continuations" is that we aren't actually using this mapping! We are constructing a different (partial, injective) ring homomorphism from arithmetic functions to meromorphic complex functions.

The idea behind this is that meromorphic functions are rather restricted in what they can do. In particular, there is at most one meromorphic function A with A(s) = \sum_{n >= 1} a(n) n^-s for s with Re(s) > k, for some k. We define our mapping from sequences to functions by mapping the sequence a(n) to the meromorphic function A(s) with A(s) = \sum_{n >= 1} a(n) n^-s for Re(s) > k for some k, if this function exists.

By the same argument as above, this is a ring homomorphism, and since it is injective we can still use information about the functions to gain information about the underlying sequences.

For example, the Riemann zeta function is not really defined by the equation Zeta(s) = \sum_{n >= 1} n^-s. It is defined to be the unique meromorphic function with Zeta(s) = \sum_{n >= 1} n^-s for all s with R(s) > 1. In particular, Zeta(-1) has nothing to do with \sum_{n >= 1} n. The latter expression doesn't define a complex number at all, but for the former it is not too difficult to show that Zeta(-1) = -1/12.

The main advantage, though, is that meromorphic functions are very well behaved. This allows us to use Cauchy's residue theorem and Mellin transforms to get very deep results about the underlying sequences. If you play this game with the "von Mangoldt" sequence you can, for instance, derive an asymptotic bound on the density of the prime numbers. This is a surprisingly simple derivation, given that this problem had the worlds greatest mathematicians stumped for a hundred years!

Summing up, the mapping or "continuation" you use is choosen (!) so that multiplication of functions corresponds to your chosen multiplication in the ring of sequences and so that you get functions which are as well-behaved as possible. There is a large design space here, and you can find different "analytic continuations" for a given sequence.