In this sentence I cannot understand the use of could. What does "could" mean here?
"The other chipmunks didn’t understand why he could like it so much; they even laughed at him!"
If I omit "could" will it change the meaning of the sentence?
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You couldn't just omit could, but you could change could like to liked.
The difference is in the basic meaning of could - potentiality.
They didn't understand why he liked it so much
means they didn't understand the reasons for the fact that he liked it.
They didn't understan d why he could like it so much
means that they didn't understand the possibility: they didn't understand how it was possible for him to like it so much (or, perhaps, how it was possible for anybody to like it so much).
It suggests that they thought that liking it so much was not just a thing that somebody might or might not do, but something impossible.