Details of the k-means++ algorithm that is used to seed k-means
You want to spread out the initial clusters, the article argues that this gives (on average) faster convergence and lower error. This means that points with a higher distance to the closest center are likely to be better candidates for initial clusters. So the probability distribution for a new initial clustering point is:

$\mathbb{P}(C = c) = \frac{D(c)^2}{\sum_{\forall x\in X}D(x)^2}$

Where $X$ is the collection of all candidate points and $C$ the chosen point distribution.

This means the higher the distance, the higher the chance it will be picked as initial point.