LORD ILLINGWORTH: The intellect is not a serious thing, and never has been. It is an instrument on which one plays, that is all. The only serious form of intellect I know is the British intellect. And on the British intellect the illiterates play the drum.
He is saying that the intellect is LIKE a musical instrument on which one plays. (Think piano, violin, clarinet, and drum) So according to your "class" and intellect, the more complex and finer the instrument, the more intelligent (better) you are.
He then says that the illiterates (those who don't read or are ill-informed or lower-class people) would be the drum players. He sees playing the drum as a simple gesture.
(This entire quote is meant for us to get an idea about Lord Illingworth's personality. He is a snob and has a low opinion of others.)