It means to try something, to 'go for it'. It requires you to be brave and do something you're scared to do. Normally it's used in a romantic sense, like when you like someone and you "shoot your shot" (take a chance) with them. "If you lilke him so much, you should shoot your shot and ask him out." "I'll shoot my shot after the meeting and ask for a raise."

Anna is correct. It is a slang phrase for taking a chance or making an attempt, usually in talking to someone that a person has romantic interest in.

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