Jazz gives a great answer here. I would add that 'many a' is an idiom. It is important to distinguish idioms from the regular meaning of words, as even native speakers who are unfamiliar with the idiom may have trouble deducing what it means.
There is a dictionary which has a section on idioms. In this case, it correctly points out that this idiom is always followed by a singular noun.
In my experience, this idiom is used for style when narrating a story.