Easy explanation of analytic continuation

Today, as I was flipping through my copy of Higher Algebra by Barnard and Child, I came across a theorem which said,

The series $$ 1+\frac{1}{2^p} +\frac{1}{3^p}+...$$ diverges for $p\leq 1$ and converges for $p>1$.

But later I found out that the zeta function is defined for all complex values other than 1. Now I know that Riemann analytically continued this function to fit all complex values, but how do I explain, to a layman, that $\zeta(0)=1+1+1+...=-\frac{1}{2}$?

The Wiki articles on these topics go way over my head. I'd appreciate it if someone can explain it to me what analytic continuation actually is, and which functions can be analytically continued?

Edit

If the function diverges for $p\leq1$, how is WolframAlpha able to compute $\zeta(1/5)$? Shouldn't it give out infinity as the answer?

I'll give you the world's simplest example. $1+x+x^2+\dots$ converges for $|x|\lt1$ only. The function $1/(1-x)$ is analytic everywhere except for a pole at $x=1$, and agrees with $1+x+x^2+\dots$ everywhere the latter is defined, so $1/(1-x)$ is the analytic continuation of $1+x+x^2+\dots$. In that sense, $1+2+4+\dots=-1$.

If all you're interested in is an explanation to a layman of why a thing like analytic continuation makes any sense to begin with (in particular, why it spits out a unique answer), the answer is the identity theorem for holomorphic functions, which in its stronger form says that if two holomorphic functions $f, g$ defined on a connected open subset $U$ of $\mathbb{C}$ are equal on a set of points in $U$ with an accumulation point (in particular any open subset of $U$), then in fact $f = g$. This is an extremely strong rigidity theorem generalizing the corresponding fact for polynomials, and shows that if you want to extend a holomorphic function $f$ defined on some domain $U$ to a function $\tilde{f}$ defined on some larger connected domain $V$, then there is at most one way to do it (since any two extensions agree on $U$ and hence agree on $V$).

One sense of "analytic continuation" is that it refers to any function $\tilde{f}$ with the above property, and another sense of "analytic continuation" is that it refers to methods for defining $\tilde{f}$ given $f$.

Sometimes it is the case that a function $f$ can be analytically continued to a very large domain $U$ but that a series or integral defining $f$ can't be made to converge on all of $U$. The philosophical point to take away from this is that holomorphic functions have a sort of "Platonic reality" that isn't necessarily perfectly captured by any particular series or integral definition of them, which are just imperfect "shadows" of this Platonic reality. The slogan I use in this situation and others like it is that

convergence is overrated.

In response to the edit, WolframAlpha is using "shadows" other than the standard series definition of the Riemann zeta function.

Answer to the added paragraph only. I guess at least this part is simple. In the case when the argument of the zeta function is a real $x>1$ the zeta function can be expressed as $$\zeta (x)=\sum_{k=1}^{\infty }\frac{1}{k^{x}}=\frac{1}{1-2^{1-x}}\sum_{k=1}^{\infty }\frac{(-1)^{k-1}}{k^{x}},\qquad(\ast)$$

where the last series is the Dirichlet eta function. Since for $x>0$ this alternating series converges (because $1/k^x\to 0$), and $1-2^{1-x}\neq 0$ for $x\in]0,1[\cup]1,\infty[$, the function $\zeta (x)$ can be continued analytically to $x\in]0,1[\cup]1,\infty[$ (note that $1$ is excluded as in the first series) as

$$\zeta (x)=\frac{1}{1-2^{1-x}}\sum_{k=1}^{\infty }\frac{(-1)^{k-1}}{k^{x}}.$$

For $x=1/2$, we get $\sum_{k=1}^{\infty }\frac{(-1)^{k-1}}{k^{1/2}}\approx 0.6049$ and

$$\zeta (1/2)=-\left( 1+\sqrt{2}\right) \sum_{k=1}^{\infty }\frac{(-1)^{k-1}}{k^{1/2}}\approx -1.4604.$$

Added: Derivation of $(\ast)$:

$$\begin{eqnarray*} \zeta (x) &=&\sum_{k=1}^{\infty }\frac{1}{k^{x}}=\sum_{k=1}^{\infty }\frac{% (-1)^{k-1}}{k^{x}}+2\sum_{k=1}^{\infty }\frac{1}{\left( 2k\right) ^{x}} \\ &=&\sum_{k=1}^{\infty }\frac{(-1)^{k-1}}{k^{x}}+2^{1-x}\zeta (x). \end{eqnarray*}$$

The relation is as follows.