Theory:

     I was smarting under the patronizing tone of Satish's mother, who obviously thought mine a very poor family: and I did not intend giving the other woman away. I let her take my hand in hers, but I could think of nothing to say. I was conscious of Satish's mother staring at us with hard, beady eyes, and I found myself hating her with a firm, unreasoning hate. The guard walked up the platform, blowing his whistle for the train to leave. I looked straight into the eyes of the woman who held my hand, and she smiled in a gentle understanding way. I leaned out of the window then, and put my lips to her cheek, and kissed her.
 
     The carriage jolted forward, and she drew her hand away.

     'Goodbye, mother!’ said Satish, as the train began to move slowly out of the station. Satish and his mother waved to each other.
Explanation:
 
Arun was feeling bad when Satish's mother was speaking with him in a dominating voice. She thought that Arun belonged to a poor family by looking at the appearance and simplicity of the woman near him. But Arun did not make the woman feel down. He was standing quietly and allowed the woman to hold his hand without any hesitation.

Arun was aware of what was going on around him. Satish's mother was already staring at both of them (Arun and the woman). Arun developed a dislike for Satish's mother without any reason. When it was time for the train to depart, the guard walked up the platform, took the whistle, and began blowing it.
 
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A guard blowing the whistle on platform

Arun looked at the woman's eyes in an understanding way (the bonding between Arun and the woman is seen). Arun leaned out of the window and gave a kiss on her cheeks.

The carriage started to move, and so the woman took her hand out. Satish bid farewell to his mother. The train started to move slowly out of the station. Satish and his mother waved their hands.
 
Meanings of the difficult words:
 
S.No
Words
Meanings
1
Patronizing Showing the attitude of superiority
2
Smarting To feel upset and angry because of failure or criticism
3
Unreasoning Not guided by or based on good sense; illogical
4
Jolted Moved with a sudden jerk
5
Waved Move one's hand to and fro in greeting or as a signal
6
Carriage Any of the separate sections of a train that carry passengers
7
Obviously In a way that is easily perceived or understood; clearly
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.