Arcana Coelestia # 5921
5921. 'That with my mouth I am speaking to you' means a revelation. This is clear from the consideration that the first thing attesting to it was the fact that they saw with their own eyes, the second that Benjamin saw with his eyes, and now the third that with his own mouth he was speaking to them. By the use of such evidence he removed all doubt that he was Joseph; he had therefore fully revealed who he was, which is why the words under explanation here imply a revelation.
Reach
The hand in the Bible represents power, which is easy to understand, so to reach out or stretch out the hand means to exercise power, to put it to work. This is an image that's used a great deal in the stories of Moses, who repeatedly stretched out his hand (or his rod, which represents a more external kind of power) to bring plagues on Egypt and perform other miracles. This is a correspondence that makes sense, since we reach out constantly to use our hands to do our work.