Ever heard of these guys? Fedor Emelianenko and Shinya Aoki? They both grew up practising Judo; it formed the base for their elite MMA careers.
So how does Judo prepare a person for mixed martial arts? The fast hand movements of Judo and angles used in kumi-kata (gripping) are very similiar to boxing. The level changes and distance closing involved in excuting throws in Judo are more diverse than wrestling. Judo also gives you a wide variety of different pins and submissions.
Perhaps the BEST thing Judo training can do for MMA is mastering the transitions. Executing a perfect throw and landing in an armbar submission is a very slick way to win a fight!