What is the connection between divergence of the correlation length and scale invariance. Why is a system scale invariant at its critical point if the correlation length tends to infinity? Is it true that the correlation length describe the typical length of the connected range?
As you say, the correlation length, $\xi$, is a measure of domain size. Two spins that are within a correlation radius will have similar statistics.
A system where there is such a scale can not be scale invariant. Indeed, scale invariance means that you can zoom in or out of the system and finds that it still looks the same. When there is a correlation length, zooming out makes it shrink and zooming in makes it expand. There is no way that you will not notice this.
The only way out is either if $\xi=0$ or if $\xi = \infty$. Then in both cases doubling or dividing the correlation length by a scale factor returns the same value of $\xi$. The case $\xi = 0$ is trivial. It means that the spins are all independent from each other. I.e. that your theory is not interacting. We are left with the critical case: $\xi = \infty$.