Many a is a somewhat archaic or poetic or literary way of saying many.
Many times I had seen her in my dreams.
Note that it takes a singular complement:
Many a time I had seen her in my dreams.
Just to throw another stick of wood onto this fire, there's another similar formulation frequently seen in literature:
Many's the time I've seen her in my dreams.
And I'll also add a quote from Shakespeare (The Merchant of Venice):
Shylock: Signior Antonio, many a time and oft / In the Rialto you have
rated me / About my moneys and my
The question this raises is, did Shakespeare recognize a difference between "many a time" and "oft"? Did it mean something more to him than how we would see it today, as interchangeable with "often"? Or was he gratuitously padding out a line to fit the meter?