As the Washington Post reports:
Acting as his own attorney, Pellicano made his closing arguments.... [He] addressed his jury. "Hi," he said. "This is the first time I will be able to speak for Mr. Pellicano." Because of the rules of the court, Pellicano the advocate must refer to Pellicano the defendant in the third person. You are correct: It is weird.
Well, it's weird by comparison to how people talk about themselves, but it's not weird by comparison to how lawyers talk about the defendant. To a lawyer, a closing argument that's chock full of "I"'s would sound very strange, and I take it the court's concern is that it will unduly personalize the process for the jury. So it's a slightly silly affectation in normal life, but legally compulsory in this unusual context.
Thanks to Language Log for the pointer.