The volume of space within the container (that must be filled with steam) is less with 6 eggs (258 ml) than with 1 egg (43 ml).
Therefore you must produce more steam (thus need more water) for 1 egg than for 6 eggs to have the same effect.
Sample:
The container, after the water and egg holder, has a volume of 500ml.
Each egg has a volume of 50 ml.
For 1 egg, 450ml of steam is needed (50+450=500ml)
For 3 eggs, 350ml of steam is needed (150+350=500ml)
For 6 eggs, 200ml of steam is needed (300+200=500ml)
The water needed for 200ml of steam (for 6 eggs) is less than the water needed for 450ml of steam (for 1 egg).