AC Machine Stator Winding
Don't know what the protocol is on answering your own questions, but it has been a few days and I have done my research, and thought it might be useful if people search for this topic in the future. 



Looking at the above equation used \$\beta = \frac{180}{\text{slots/pole}} = 30º\$. The significance of beta is the angular displacement between successive slots. This might be difficult to visualize so I drew up the following winding diagram for a 3 Phase AC machine with 24 slots and 4 poles using the table shown in the question:





Looking at the above diagram \$\beta\$ is essentially the distance between slots 1-2,2-3,3-4...

Therefore one coil, such as coil 1 which traverses slot 1 and 7 falls under the north and south pole exactly because they are 180º apart as seen in the image above.

Now labeling the starting end of each coil is easy, assuming an RYB phase sequence taking red as the reference. We will start Red at slot 1. Looking at the diagram in the question we follow the current direction and end at slot 19.

The yellow phase is displaced by 120º and looking at the winding diagram, a 120º displacement from slot 1 is slot 5. Hence we start the yellow phase at slot 5 and continue as seen in the question.


  The blue phase is now easy to understand because it is displaced by 240º. A 240º displacement from slot 1 leads to slot 9 and hence we start at slot 9.