Theory:

     'Well, be very careful Arun,' said Satish's mother looking sternly at me through her spectacles. 'Be very careful when your mother is not with you, and never talk to strangers!'
 
     I looked from Satish's mother to the woman who had given me tea and sweets, and then back at Satish's mother.

     'I like strangers,’ I said.

     Satish's mother definitely staggered a little, as obviously she was not used to being contradicted by small boys. 'There you are, you see! If you don't watch over them all the time, they'll walk straight into trouble. Always listen to what your mother tells you,’ she said wagging a fat little finger at me. 'And never, never talk to strangers.'
Explanation:
 
Satish's mother, through her spectacles, looked at Arun seriously and told him to be very careful when he was not with his mother and to never speak to strangers. Satish's mother did not know the woman standing near Arun was a stranger, but also she warned him strictly to be aware of the strangers and never talk to them. Arun looked back at Satish's mother after looking at the woman who had bought him tea and snacks.

Arun answered as he like strangers. The reason behind Arun's statement was the woman standing nearby him was very kind and caring as she had bought him tea and snacks when he was alone on the railway platform. She reduced his appetite, and also, she spoke to him in a gentle way; Satish's mother was shocked because she had not expected that answer from Arun. Later, she disagreed with him, warning him that he would be in trouble if he did not follow others and listen to their words.

Satish's mother never experienced arguing with small boys. She always had the thought that the children would listen to her speech. But here, Arun was entirely against her view. Satish's mother was a strict lady. She had a mindset of the kids need to follow her words. She was very talkative and stubborn in her ideas. As a tough lady, she pointed out her fat finger and told him to obey his mother.
 
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Satish's mother pointed out her fat fingers towards Arun
 
Meanings of the difficult words:
 
S.No
Words
Meanings
1
Sternly In a serious and severe manner, especially when asserting authority or exercising discipline
2
SpectaclesA pair of glasses
3
Staggered Walked or moved unsteadily as if about to fall
4
Wagging Move rapidly to and fro
5
Contradicted Deny the truth of a statement by asserting the opposite
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.