Theory:

     "Good-bye," I said to the other woman, "Goodbye — mother …"

     I didn't wave or shout, but sat still in front of the window, gazing at the woman on the platform. Satish's mother was talking to her, but she didn't appear to be listening; she was looking at me, as the train took me away. She stood there on the busy platform, a pale sweet woman in white, and I watched her until she was lost in the milling crowd.
Explanation:
 
Before leaving the station, Arun said goodbye to the woman and called her as 'mother'. Arun's calling the woman as 'mother' at the time of parting was a sweet gesture of recognising a loving relationship between the boy and the woman. While the train started to move, Arun did not wave his hands or shout with excitement. He sat near the window seat, staring at the woman standing on the platform. Satish's mother was constantly speaking to her, but she was not paying attention. Furthermore, she was only looking at Arun. The train slowly started to move. The woman stood on the platform in her white sari and he watched her until the train took him away from the station. In his view, while the train moved on, she disappeared into the large crowd.
 
Meanings of the difficult words:
 
S.No
Words
Meanings
1
Milling crowdAn aggregation of individuals, usually gathered in a public area
2
Gazing Look steadily and intently, especially in admiration, surprise, or thought
Reference:
State Council of Educational Research and Training (2019). Term-3 English Standard-8. The Woman on Platform 8 - Ruskin Bond (pp. 107-114). Published by the Tamil Nadu Textbook and Educational Services Corporation.