Theory:

The old lady was sitting in a wheelchair, her hands folded in her lap. She had silver white hair pinned into a wispy bun. She was gazing out at the garden. “Hello,” I said. She turned and looked up at me vacantly. “Happy Christmas, Connie,” I went on. “I found this. I think it’s yours.” As I was speaking her eyes never left my face. I opened the tin box and gave it to her. That was the moment her eyes lit up with recognition and her face became suffused with a sudden glow of happiness. I explained about the desk, about how I had found it, but I don't think she was listening. For a while she said nothing, but stroked the letter tenderly with her fingertips.

Suddenly she reached out and took my hand. Her eyes were filled with tears. “You told me you’d come home by Christmas, dearest,” she said. “And here you are, the best Christmas present in the world. Come closer, Jim dear, sit down.”

I sat down beside her, and she kissed my cheek. “I read your letter so often Jim, every day. I wanted to hear your voice in my head. It always made me feel you were with me. And now you are. Now you’re back you can read it to me yourself. Would you do that for me, Jim dear? I just want to hear your voice again. I’d love that so much. And then perhaps we’ll have some tea. I’ve made you a nice Christmas cake, marzipan all around. I know how much you love marzipan.”
Explanation:
  
The author found Mrs. Macpherson was sitting in a wheelchair with her hands in her lap, looking into the garden. He saw that her white hair neatly pinned into a fine bun. He said hello and wished her Happy Christmas. He continued that he found that letter in the desk and told that it was her letter. Immediately she became bright with excitement. He opened the tin box and handed over the letter to her. Happiness spread gradually all over her face. He explained about how he got the letter, how he got the desk, etc but he understood that she was not listening to him. She did not say anything for some time, but she was caressing the letter fondly with her fingertips.
 
Suddenly she came forward and took my hands. Her eyes were filled with tears. She told him that, he had told that he would come home for Christmas. She continued and told him that he was the best Christmas present in the world. She mentioned him as "Jim dear" and asked him to sit down. She had thought it was her husband, Jim Macpherson, who had come home. She had been mentally disturbed - may be due to age, loneliness, fire accident, missing her husband or any other reason. So for Connie, his return was the best Christmas present in the world.
 
The author sat down beside her, and she kissed him on the cheek. She told him that she read her letter every day and wanted to hear his voice in his head, as it always felt as if he was near her. All along, she assumed she was talking to her husband, Jim. Further, she asked him to read the letter to her so that she could hear his voice. She continued and told that she would love it if he read the letter. Then she made plans of having tea with him. She told him that she had made him a nice Christmas cake with marzipan all around, as she knew that he loved marzipan very much.
 
It is seen that the author chose not to reveal his identity. By doing so, he is only going to disappoint an old lady. So he chose to keep her happy, by letting her think he was indeed Jim Macpherson.
  
  
Meanings of difficult words:
  
Words
Meanings
Wispyfine, feathery
Vacantly(here) blank, vague look
Lit upbecame bright with excitement or happiness
Suffused withspread (happiness all over) gradually
Strokeuse hands gently on a surface, caress
 
Reference:
National Council of Educational Research and Training (2008). Honeydew. The best Christmas present in the world - Michael Morpurgo (pp. 9-16). Published at the Publication Division by the Secretary, National Council of Educational Research and Training, Sri Aurobindo Marg, New Delhi.