I may be misreading but I'm inferring from the text that Latin drills weren't useless, hence the quote marks. Is it not implying that he was getting taught something that would later become relevant?
I'm just confused about who would even suggest Latin was "useless" in the first place. But maybe something is missing in the quote.
The author is pointing out that there was no audience or need for the type of linguistic creativity that Shakespeare was only able to achieve as a result of his intimate knowledge of Latin, until the professional theater emerged during Shakespeare’s very lifetime. The drilling and study in Latin proved, in the event, to be practical and useful in a new and unexpected way.