When he comes to after fainting, what does the narrator remember?

The answer to your question is found a full paragraph after the narrator tells his readers that he had "swooned."  The reason for that is because the narrator begins philosophizing with the reader about how the body slowly comes to be more alert after fainting.  This is the paragraph that really taxes most of my students.  But after that paragraph the narrator finally gets around to telling his readers about a faint memory.  The memory is of being lifted up and being carried down.  Down a long way.  The narrator remembers that he felt the descent was so long that he became dizzy from it.  At the very end of his descent and memory, the narrator then tells his readers that he remembers flatness and dampness.  The flat damp location is his cell with the pit and the pendulum.  

These shadows of memory tell, indistinctly, of tall figures that lifted and bore me in silence down—down—still down—till a hideous dizziness oppressed me at the mere idea of the interminableness of the descent. . . After this I call to mind flatness and dampness.