Theory:

     We found eleven. We fixed six on one side and five on the other, and half an hour later the wheel was in its place again. It need hardly be added that it really did wobble now; a child might have noticed it. He said it would do for the present.
 
     I said, “Watching you do this is of real use to me. It is not only your skill that fascinates me, it is your cheery confidence in yourself, your inexplicable hopefulness, that does me good.”
 
     Thus encouraged, he set to work to refix the gear-case. He stood the bicycle against the house, and worked from the off side. Then he stood it against a tree, and worked from the on side. Then I held it for him, while he lay on the ground with his head between the wheels, and worked at it from below, and dropped oil upon himself. Then he took it away from me, and doubled himself across it till he lost his balance and slid over on to his head.
Explanation:
 
After looking for the missing balls, the men were able to find about eleven of them. They fixed six of them on one side and the rest on the other side. It took about half an hour for them to get the wheel in its place. Unfortunately, the wheel that never wobbled earlier had started to wobble then. But the man didn't seem to be affected with newly-found wobbling. He said it was fine and could be managed for the time being.
 
The narrator commented on the cheerful manner of the man. He said that the man's confidence and hopefulness even during times when it is least called for fascinates him. While the narrator had no intention to bring in a sarcasm here, it is evident to the readers that the man's action and the narrator's reaction deserved to be mocked at. However, the narrator is a kind person, and he knows not how to mock a person.

The narrator's words encouraged the man to work on the gear-case. He sat at his work with a renewed energy. He picked the bicycle and positioned it against the wall of the house so that he could work on the right side of the bicycle.
 
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The man tried to fix the gear-case

Later, he put the cycle against a tree and worked on its right side. After a while, the man asked the narrator to hold the bicycle so that he could work from below. He lied on the ground and put his head between the wheels. While working, he dropped oil upon himself.
 
When none of the postures worked, the man took the cycle from the narrator and decided to fix it all by himself. He positioned the cycle against himself and bent over the other side of the cycle. Unfortunately, he lost his balance, tripped over, rolled on his head, and reached the other side of the cycle.
 
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The man doubled himself across the bicycle
 
Meanings of difficult words from the paragraphs:
 
Sl. No.
Words
Meanings
1
FascinatesAttract the strong attention and interest of someone
2
Inexplicable
Unable to be explained or accounted for
3
HopefulnessThe feeling or state of having hope
4
Refix
Fix something again
Reference:
National Council of Educational Research and Training (2007). Honeycomb. A Bicycle in Good Repair (pp. 126-132). Published at the Publication Division by the Secretary, National Council of Educational Research and Training, Sri Aurobindo Marg, New Delhi.