It affects the amount of fleet supply you can have at a given tax rate and also the maximum fleet supply.  Every ship consumes a certain amount of fleet supply.  For example, an Advent scout consumes 2 supply points.  A heavy fighting cruiser, the Advent's Destra Crusader, consumes 10 fleet supply points.  So having 5 scout ships = 1 heavy cruiser in terms of fleet supply.

At the Normal fleet supply setting you start out with 100 fleet supply, so you will soon need to research the first fleet supply increase to go up to 250 points and a 9% tax on your credits and mineral extractions.  It goes up to a maximum of 2000 fleet supply and a prohibitive 75% tax.  The small setting gives you less fleet supply at each tax rate and a smaller maximum while the large setting gives you more and a higher maximum.