This is part of the poem's 'Journey to Emmaus' section, but in the notes Eliot explained: The following lines were stimulated by the account of one of the Antarctic expeditions (I forget which, but I think one of Shackleton's): it was related that the party of explorers, at the extremity of their strength, had the constant delusion that there was one more member than could actually be counted.
It turns out that a fair number of people in extremely stressful conditions, often explorers, report an encounter with a mysterious "Other." Unlike sleep paralysis or experiences of "hauntings," these presences seem supportive or helpful and have given some people the determination to carry on and survive harrowing conditions. It's not always a get-out-of-jail free card - Geiger cites one case where a climber described the experience in his diary before dying. The book uses some neurological findings to argue that the Other is a psychological survival mechanism, a hallucination triggered by the brain in certain cases of stress.
Geiger cites some people who interpret this in a dismissive fashion - standard-issue "See, all this mystical bunk is just in your head!" rhetoric. He also quotes those who believe that something supernatural or paranormal is being encountered, beyond how the brain interprets it. The introduction to the book is from Vincent Lam, the Canadian doctor and award-winning writer. Lam is an Anglican (the last I heard anyways) and in his introduction he describes a stressful period in his life when he encountered what he believes was "a guardian angel."Geiger himself takes a stance between these positions. He seems to conclude that it's "just" happening in the brain, but that it's a wondrous, even spiritual experience at the same time. For him, it speaks to human inter-connectedness, our need for others, and the amazing flexibility of the human psyche. There's an interview with Geiger here. The Globe and Mail reviewed the book here.
On a related note, have you seen Foster's Home for Imaginary Friends?