I object to rows because my nerves are shaken,
As far as I know, "row" means "a straight line or something similar to that" but I guess that definition doesn't fit into the category in this case.
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Your definition is correct - a row is a line of things, people, animals, etc. arranged next to each other. This definition of the word is pronounced /roʊ/ (rhymes with 'toe'.)
However, row also has another definition - it can also mean to have a noisy quarrel or argument. For example:
My parents often have rows, but my dad does most of the shouting.
From the sentence you have provided, the speaker is already quite on edge, therefore noisy arguments would not be something he or she would like to be a part of - therefore they object to, or are against them. Although spelt the same as the previous definition, it's not pronounced the same - i.e. it's a homograph. This definition of the word is pronounced /raʊ/ (rhymes with 'how'.)