Imagine that you have a container of potential limitless capacity. The container starts with smalls capacity, equal to the real limited capacity that the real algorithm uses. As you add elements, when the container is full, its capacity is doubled, but all elements are then placed in a random position.
When you're done, you're told the occupancy of the subset of the large container corresponding to the initial size and how many times the container size was doubled. Multiplying that occupancy by the power of two of the number of doubling gives you an approximation of the real size.
The small catch is that in the actual algorithm, due to the discarding, the final number of elements, the occupancy, is somewhat erroneous.
EDIT
Another way to say this: you got a container of limited capacity S. When full, you "virtually" double its size and then randomly move elements over the full "imaginary" size of the virtual container. So after the first filling, you end up with about 1/2 the elements. After the second filling 1/4, etc. Also, since now your "virtual" container is larger, when you add a new element, there is only 1/2^n the it will be place inside your limited-capacity view of the entire virtual container.
At the end, the approximate real count is the number of elements you got multiplied by 2 to the power of the number of size doubling.
Again, it is as if you have a small window into a limitless container.