Many a failure..... DOES IT MEAN THE SAME AS Many failures? If so, why do we say "many a something" instead of "many something"?

By the way, what does it mean: And many a failure turns about, when he might have won had he stuck it out.
Yes it means the same thing. Many a is a slightly dated or literary way of saying it.

There's many a slip 'twixt the cup and the lip.
After many a summer dies the swan.

and from the Columbia Guide:

many a
is a locution that is followed by a singular noun, which in turn takes a singular verb: Full many a flower is born to blush unseen … (Thomas Gray, Elegy Written in a Country Churchyard)