Vincent Spaulding wants to get Mr. Wilson out of the way for a few hours a day so he can use the basement of his shop to tunnel into the bank next door.
Vincent Spaulding’s real name is John Clay. He is an accomplished criminal, and Holmes seems to think he is pretty smart.
“He is, in my judgment, the fourth smartest man in London, and for daring I am not sure that he has not a claim to be third. I have known something of him before.” (etext pdf p. 13)
Clay wants to rob the bank, so he devises a plan to trick Mr. Wilson into leaving each day for several hours. He gets himself hired as his assistant, and then shows him an advertisement for the “Red-headed League.” Clay applies and is accepted, of course, and just copies the Encyclopedia Britannica. Meanwhile, Clay digs a tunnel into the bank next door.
It would have worked if they had not closed up shop suddenly, causing Mr. Wilson to get suspicious and consult Holmes. Holmes knew something was up, and quickly solved the case.