Breaking a man up into head, torso, shorts, legs, shoes, we see that the following bits go missing:
The 13-man configuration is shown first with the 12-man one below it. Take the 13-man configuration as the starting point. Number the men as 1, 2, 3 from the front in the first diagonal row, 4, 5, 6 in the second, 7 and 8 in the third, 9, 10, 11 in the fourth, and 12, 13 in the last. There is no man at position #6 after the move.
Breaking a man up into head, torso, shorts, legs, shoes, we see that the following bits go missing:
a) Man #1 is missing his scalp, #4 has a small head that becomes large after the move, #12 has only half a head (the top of his singlet doubles as a chin). This explains the loss of one head.
b) The white shirt of #5 gets added to that of #10 whereas the top part of the shirt of #10 becomes the long upper chest of #2. This accounts for the loss of one white shirt.
c) The very brief black shorts of #13 merge with the shorts of #5 making a long pair. This explains the loss of one pair of black shorts.
d) The extremely short legs of #8 get added to those of #13, explaining the loss of one pair of legs.
e) Men #3, 6 and 11 have incomplete black shoes. The shoes of #3 and #8 are combined, as are those of #6 and #11.
Even after seeing all these individual sleights of hand it is hard to see how it all hangs together. What makes it difficult is that so many bits are recombined to dis-assemble the 13th man.