There are a few different ideas or theories about why people believed that President Lincoln asked to see John W. Booth on the night of his assassination.
One theory is that this rumor may have started because Booth was known to be a well-known actor, and Lincoln was said to have enjoyed attending the theater. Another theory suggests that some people may have believed that Lincoln and Booth knew each other personally, although there is no evidence to support this claim. It's also worth pointing out that there were many rumors and false stories circulating in the aftermath of Lincoln's assassination, and it's possible that the idea that Lincoln had requested to see Booth was simply one of many unsubstantiated claims.