During the early Eocene, the continental configuration was such that the Arctic sea was almost entirely cut off from the wider oceans.  This meant that mixing — provided today by deep water currents such as the Gulf Stream — did not occur, leading to a stratified water column resembling today's Black Sea.
High temperatures and winds led to high evaporation, increasing the density of the ocean, and — through an increase in rainfall — high discharge from rivers which fed the basin.  This low-density freshwater formed a nepheloid layer, floating on the surface of the dense sea.
Even a few centimetres of fresh water would be enough to allow colonization by Azolla; further, this river water would be rich in minerals such as phosphorus, which it would accumulate from mud and rocks it interacted with as it crossed the continents. To further aid the growth of the plant, concentrations of carbon (in the form of carbon dioxide) in the atmosphere are known to have been high at this time.
Blooms alone are not enough to have any geological impact; to permanently draw down CO and cause climate change, the carbon must be sequestered by the plants being buried and the remains rendered inaccessible to decomposing organisms.  The anoxic bottom of the Arctic basin, a result of the stratified water column, permitted just this; the anoxic environment inhibits the activity of decomposing organisms and allows the plants to sit unrotted until they are buried by sediment.