What does "a bat to clear the air"mean? Why use it?
Bat
For juicy June bugs, meaty moths, mosquitoes.
They are scary. But there is nothing like a bat to clear the air?

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yes. i second the answer "to eat them". 
side note: to clear the air can also mean to address an issue you have with someone. for example, Last week, Jill insulted Jim at dinner. Jill wasn't sure if Jim was still angry, so Jill decided to clear the air with Jim.

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It means that a bat is a good predator of flying insects ("to clear the air of mosquitoes" :-)

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I'm not sure what this is supposed to mean. Can you give us some context?