“Please don’t go away”, she said.

The correct answer is D. She begged me not to go away.

A. She said to please her and not go away.
This is

B. She told me not to go away.
This is not the correct answer because the speaker is not telling the listener not

C. She begged that I not go away.
This is not the correct answer because the speaker is not begging that the listener not go away. She is begging the listener not to go away.

D. She begged me not to go away.
This is the correct answer because the speaker is begging the listener not to go away. She is using a more direct and emotional tone than in the other options.