The word "mouth" here represents Aaron repeating what Moses told him. Alternate translation: "He will say what you tell him to say" (See: [[rc://en/ta/man/jit/figs-metonymy]])
The word "like" here means Moses would represent the same authority to Aaron as God did to Moses. Alternate translation: "you will speak to Aaron with the same authority with which I spoke to you" (See: [[rc://en/ta/man/jit/figs-simile]])