House was trying to win a bet with Wilson by proving Kayla's asexuality was the result of a medical condition. However, when that turned out to be fruitless, he called Randy in on a pretext of giving him a flu shot and gave him a full battery of tests. Wilson protested that House promissed not to have any contact with the patient, but House reminded him that the deal was that he wouldn't have any contact with Kayla and nothing was said about Randy.
Randy's tests showed he had a slow growing brain tumor. It was placed against the glands that control sexual desire, and had probably been affecting him since puberty. Wilson broke the news to Randy, who resisted treatment when he was told that it would increase his sex drive. However, Kayla insisted on treatment. Randy said that he thought sex was disgusting and he didn't want to put Kayla through it. However, she broke the news to him that she had had sex before and it wasn't that bad - it was actually kind of fun.
Wilson realized House was right - Kayla did have sexual interests and suppressed them because Randy didn't want sex. He paid off the bet.
In British English, "randy" means the equivalent of the American English "horny". For a similar reference, see Willie