Why does benzoic acid precipitate out when the aqueous layer is acidified with HCl?

You're starting with something like sodium benzoate (depends on what base you used for the extraction step), which contains sodium ions (Na+) and benzoate ions (C6H5COO-). Ionic compounds like that tend to be soluble in water. When you add H+, you protonate the benzoate ion to make benzoic acid (C6H5COOH), which is a neutral molecule, and hardly soluble in water at all... so it precipitates.