Theory:

     “Oh, like most of them!” I answered; “easily enough in the morning; goes a little stiffly after lunch.”
 
     He caught hold of it by the front wheel and the fork, and shook it violently.
 
     I said, “Don’t do that; you’ll hurt it.”
 
     I did not see why he should shake it; it had not done anything to him. Besides, if it wanted shaking, I was the proper person to shake it. I felt much as I should had he started whacking my dog.
 
     He said, “This front wheel wobbles.”
Explanation:
 
To the man's question as to how the bicycle runs, the narrator replies humorously. He says that the bicycle runs easily in the morning but becomes stiff in the afternoon.

The sentence spoken by the narrator might have various undertones. The explicit meaning is that the bicycle may become stiff after being used all morning. On the other hand, the narrator may be referring to the condition of humans rather than the machine. Humans have the tendency to feel lazy and tired post-lunch. The sentence can also mean that the narrator is indirectly telling the man that the man has arrived late and that it is high time that they began their journey since both the cycle and the humans may feel stiff after lunch. Hence, the line spoken by the narrator can be a source of sarcasm or might be void of any sly intentions. However, taking the personality of the narrator into account, it is highly possible that he had never intended to pass a sarcastic comment against the man.

Back at the garden, the man decides to give it a look. He holds the bicycle at the front wheel and fork. Then he shakes it violently. This upsets the narrator. He tries to stop the man through feeble protests.
 
Bikefork.jpg
Bicycle fork

Later, the man declares that the front wheel of the bicycle wobbles.
 
Meanings of difficult words from the paragraphs:
 
Sl. No.
Words
Meanings
1
StifflyIn a manner that is not relaxed or friendly
2
WhackingTo strike forcefully with a sharp blow
3
WobbleTo move unsteadily from side to side
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.
 
pc: https://commons.wikimedia.org/wiki/File:Bikefork.jpg