Theory:

     And so we walked on. Gradually I discovered that I was ten times more thoughtful than I ever thought I could be. I also realised that if I had to describe people and things to someone else, it made them more interesting to me. When I finally had to leave, I told Miss Beam that I was very sorry to go. "Ah!" she replied, “then there is something in my system after all.”
Explanation:
 
The day turned out to be better than the narrator had expected. The narrator realised that he had become ten times more thoughtful than he ever thought he could be. His conversation with the girl had brought a transformation in him. He found out that he liked describing things to people, and the act made those things appear more interesting than before.

Finally, the time had come for the narrator to leave. A day in the school had taught him more than he had imagined. He met Miss Beam and said that he was feeling sad about leaving.
 
narrator sad.png
The narrator was sad because it was time for him to leave

Miss Beam was happy to learn that the narrator was pleased with the unique system.
 
happy miss beam.jpg
Miss Beam was happy to know that her system works
 
The lesson concludes by Miss Beam responding to the narrator's feedback. She says, "Ah! then there is something in my system after all.” The narrator's remark proves that he has become attached to the school and has found the system highly unique. Hence, his remark reminded her that her system is effective and worthwhile.
 
Meanings of difficult words:
  
Sl. No.
Words
Meanings
1
Gradually Slowly
2
DiscoveredRealised; found out
3
Sorry to goFeeling sad to go
4
Something in the system
The system is effective and worthwhile
Reference:
National Council of Educational Research and Training (2006). Honeysuckle. A Different Kind of School - E. V. Lucas (pp. 56-62). Published at the Publication Division by the Secretary, National Council of Educational Research and Training, Sri Aurobindo Marg, New Delhi.