Theory:

     'Hello, Arun!’ he called. 'The train's coming in late, as usual. Did you know we have a new Headmaster this year?'

     We shook hands, and then he turned to his mother and said: 'This is Arun, mother. He is one of my friends, and the best bowler in the class.’

     'l am glad to know that,' said his mother, a large imposing woman who wore spectacles. She looked at the woman who led my hand and said: 'And I suppose you're Arun's mother?'

     I opened my mouth to make some explanation, but before I could say anything the woman replied: 'Yes I am Arun's mother.'
 
     I was unable to speak a word. I looked quickly up at the woman, but she did not appear to be at all embarrassed, and was smiling at Satish’s mother.
Explanation:
 
Satish and Arun met each other at the railway station. They shook their hands and began conversing about the train's expected arrival time. Satish informed Arun that the train would be late as usual and asked, "Did you know we have a new Headmaster this year?"
 
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Arun and Satish shake their hands
 
Later on, Satish thought of introducing Arun to his mother. Satish introduced Arun as one of his friend and the best bowler in the class. Satish mother felt happy seeing Arun and said, 'I am glad to know that'. Satish mother was wearing spectacles, and she looked like a large imposing woman (means grand and impressive in appearance).
 
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Satish and his mother

Satish mother looked at Arun and the woman standing nearby him. She was holding Arun's hand, and so Satish mother asked Arun, 'And I suppose you're Arun's mother?'. Arun knew that the woman standing near him was not his mother. So he opened his mouth to give some clarification about his mother, but before that, the woman standing near Arun said, 'Yes, I am Arun's mother.'

On hearing the woman's words, Arun became speechless, and he looked at her. But she was standing simply without any discomfort and smiling at Satish mother.
 
Meanings of the difficult words:
 
S.No
Words
Meanings
1
SpectaclesA pair of glasses
2
Imposing Grand and impressive in appearance
3
EmbarrassedFelt awkward, self-conscious, or ashamed
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.