One small tank of helium can fill hundreds of balloons because helium in a tank is at a high pressure and low volume. When the helium leaves the tank and fills a balloon, the pressure is lowered and the volume of the helium increases.
The volume of a certain amount of any gas, including helium, is dependent on both pressure and temperature. Gas volume increases with lowered pressure or increased temperature, and decreases with increased pressure or lowered temperature. This behavior is described by gas laws, the simplest of which is the ideal gas law. The ideal gas law is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the universal gas constant and T is temperature.
Suppose the behavior of helium is described by the ideal gas law, and imagine holding n and T constant. Then the ideal gas law states that when the pressure is decreased, the volume of the gas increases proportionally in order to keep the product of P and V equal to nRT. This situation is exactly that of filling balloons from one tank of helium. Since the amount of helium is not changing, the change in pressure is what causes an increase in the volume occupied by the helium.