What is the size of a queue associated with a resource when the resource becomes a bottleneck?
He explicitly says it's a rule of thumb.  That means it's not a hard-and-fast rule; it is just a guideline that is often helpful (but no guarantees).  While I'm not familiar with that textbook, I suspect that justifying this rule of thumb would likely require concepts that are beyond the scope of that book. 

If you want to understand where it is coming from, one approach is to try some simulations and see what happens.  That will give you a rough feeling for it.

If you want to understand the mathematics behind it, your next step is to study queuing theory.  There is a relationship between the average queue length and the average utilization of a resource.  When queues get large, that tends to mean that the arrival rate is close to or exceeds the departure rate, and means that the resource is highly utilized.  In the kind of setting talked about in the book, this in turn means that the resource is a good candidate to be a bottleneck.