Why the requirement of the GCD of the lengths of all circuits in the graph being one?
This is my understanding: the statistics of the source described in the paper depend on which character is produced first. If the first character is $a$, then one of the source properties is that letters in odd positions are always $a$. However, if the first letter is $b$ (in other words, a shift of $1$ in the circuit), then letters in odd positions can be either $b$ or $c$. This is what makes the source non-ergodic.

Note that the statistics when choosing $b$ as first letter and $c$ as first letter are exactly the same. In other words, there are $d=2$ different classes of sequences, and those in each class have the same statistics.

Regarding the importance of these sources: for one, Shannon's theorems depend on the source being ergodic. From a more practical standpoint (and this is purely my interpretation), non-ergodic sources are not good models for real-world processes; referring back to the example, no practical language produces repeated letters in a fixed periodic pattern.

I recommend reading "An Introduction to Information Theory: Symbols, Signals and Noise" by John R. Pierce, along with Shannon's paper. It explains many of the concepts in the paper in a very didactic manner. For example, it takes four pages to very clearly explain ergoidc sources, with nice examples.