What goes inside the filter?

Nothing!  The coffee grinds begin free floating in water, figure 1, and end up below the filter separate from the dissolved solids, figure 3.  The filter, 2a/b comes apart for cleaning, and is closed without anything inside it before being used to filter the grinds.  Its shape is the way it is for a different reason:  it allows the lower filter, 2a, to punch beneath the floating coffee and filter from the bottom like a vacuum brewer, while the upper filter finishes the job.