It was simply a ball basket for convenience. For toting balls to and from the tables and even making the racking go quicker.
Thanks Mr. Bond, that makes sense, I'd never considered how people moved sets of balls around before the era of the plastic ball rack.
Two further questions -
1) How did the basket make racking go quicker?
2) Was the "Kling" model table named after Johnny Kling himself? (Seems likely from the article but it could just be coincidence)