Correlation length at low temperatures? The correlation length gives (approximately) the distance over which a spin flip has an effect. For systems with ordered phases, at low temperatures the correlation length is then small (since a single spin flip will have little affect due to the large energy needed to rotate all the spins).
But the one-dimensional Ising model (although it doesn't have an ordered phase) has a correlation length that diverges at low temperature. Given what I have said above and given at $T=0$ for the 1D Ising we do get an ordered phase, why is the correlation length diverging instead of going to zero? 
 A: I assume that the question is not about how one shows that the correlation length diverges, as this is a simple computation when $d=1$, but rather about intuition behind the result.
First, note that the same also happens when you consider the $d$-dimensional Ising model, with $d\geq 2$, as the temperature approaches $T_c$ from above. (Actually, also from below, but let's stick to the situation you are interested in.)
In both cases, the reason is the same (although the mechanism is more subtle in higher dimensions). The correlation length is the relevant length-scale in the system. So, above $T_c$, it measures, for example, the typical size of clusters of spins taking the same values. The size of these clusters diverges as the system gets closer and closer to the critical temperature.
Rather than the kind of dynamical interpretation you seem to be using, you should rather interpret the correlation length in the following statistical manner: suppose you only observe a single spin of your system (say, the spin at $0$, $\sigma_0$) and you discover that it takes the value $+1$. What can you say about the value of the spin at $i\neq 0$? Well, if $i$ is "close enough" to $0$, then knowing that you have a $+1$ at $0$ makes it more likely to observe a $+1$ also at $i$. The correlation length quantifies what "close enough" means. Namely it is the typical distance up to which the probability of observing a $+1$, given that $\sigma_0=+1$, differs significantly from $1/2$.
Now, for the one-dimensional model at low temperature, observing that $\sigma_0=+1$ makes it very likely that you'll see only $+1$ up to very large distances, precisely because the energetic cost is huge to flip spin. This will occur (the cost being finite), but the density of pairs of neighbors with spins taking different values goes to zero as $T\downarrow 0$. The average distance between two consecutive such pairs will be of the order of the correlation length, so it diverges.
Addendum
Let me stress the difference with what happens as $T\downarrow 0$ in higher dimensions. In this case, the system is in an ordered phase. For definiteness, let's assume that it is in the $+$ state $\mu^+_T$. In this case, the correlation length measures the typical distance over which
$$
\mu_T^+(\sigma_i=s \,|\, \sigma_0=s)
\text{ differs significantly from }
\mu_T^+(\sigma_i=s).
$$
And this distance decays to $0$ as $T\downarrow 0$
, for reasons explained in this answer.
