Theory:

     When I arrived there was no one in sight but a girl of about twelve. Her eyes were covered with a bandage and she was being led carefully between the flower-beds by a little boy, who was about four years younger. She stopped, and it looked like she asked him who had come. He seemed to be describing me to her. Then they passed on.
 
     Miss Beam was all that I had expected — middle-aged, full of authority, yet kindly and understanding. Her hair was beginning to turn grey, and she had the kind of plump figure that is likely to be comforting to a homesick child. I asked her some questions about her teaching methods, which I had heard were simple.
Explanation:
 
When the narrator reached the school, he found two kids walking through the school garden.
 
garden.jpeg
Garden
 
One of the kids was a girl of twelve years old. She was blindfolded with a bandage.
 
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A girl with a bandage on her eyes
 
The other kid was a little boy. He must have been about eight years of age.
 
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A young boy
 
The boy was helping out the girl by leading her carefully between the flower-beds.
 
flower beds.jpg
Flower-beds
 
The girl must have sensed the narrator's presence. She stopped walking and asked the boy something. The narrator assumed that she had asked the boy about him. The boy, in turn, described the narrator to her. Later, they moved on.
 
kids walking.jpg
The boy leads the blindfolded girl through the garden

The narrator walked further to Miss Beam's office. Miss Beam was all that the narrator had expected. She was a middle-aged lady with her hair beginning to grey. She appeared to the narrator as someone kind, understanding, and full of authority. She looked like she could be both soft and strict at the same time. Her plump figure gave a comforting look. The narrator felt that Miss Beam's appearance would make a homesick child feel better.
 
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Miss Beam in her office
 
Later, the narrator started his conversation with Miss Beam. He seemed to have questions regarding the school and its teaching methodology. The narrator had heard that the teaching methods followed by Miss Beam were simple, and hence he started talking about them.
 
This paragraph is important because we get to know more about two of the central characters of the story. Miss Beam is pictured as someone who is kind, bold, older, and plump. The narrator, on the other hand, could be seen as someone who is curious and has come to the school to inquire and learn more about it. 
 
Meanings of the difficult words:
  
Sl. No.
Words
Meanings
1
In sightTo be seen
2
LedGuide someone in their movement, mostly by holding their hands
3
Flower-bedsA garden plot in which flowers are grown
4
Passed on
Moved on; walked away
5
All that I had expectedAs imagined or expected
6
Middle-agedSomeone whose age is between 45 and 65
7
AuthorityThe power to give orders, make decisions, and enforce obedience
8
KindlyFriendly
9
PlumpStout; Pleasantly fat
10
HomesickExperiencing a longing for one's home during a period of absence from it
11
MethodPractise; system
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.