Did you notice the little-person under the second 'e' in 'Leprechaun'? In going from 14 to 15 leprechauns, the poor guy lost a knee-cap! The man under the red 'THE' got the knee-cap, but gave up a foot.
Try rearranging the leprechauns in the order of give-and-take. Start with the knee-cap-loser. Then comes the knee-cap-gainer/foot-loser. Then the chap who gained that foot, and so on. This continues to a little man with crossed arms who acquires a toupe of sorts.
In order, the 14 complete leprechauns are ...
Instead of three pieces, the above image has two pieces - top and bottom. Slide the bottom over, giving the second man the first man's knee. What do we find?
Using this ordering of the leprechauns, it is clear where the extra man comes from. Each of the original 14 men (but the last) gives some of himself to the next man, but each man receives less than he gives. The extra leprechaun pieces add up to form a new man!