He was one of those earnest, persevering dancers--the kind that have taken twelve correspondence lessons.
The quote describes a particular type of dancer known for their dedication and determination. This dancer has gone to great lengths, including taking numerous correspondence lessons, to improve their skills. The use of "earnest" and "persevering" suggests that they are serious about their passion for dance, even if their natural ability may not match their commitment.
The reference to "twelve correspondence lessons" highlights the lengths to which this individual will go to master their craft. It presents an amusing image of someone who is earnest in pursuing their dancing dreams, depicting a character who embodies both effort and perhaps a touch of earnestness that may not translate into actual talent, a recurring theme in P.G. Wodehouse's humorous storytelling style.