Here is the rule:
"CREATURES ADJACENT to a burning skeleton take 1d6 points of fire damage at the START OF their (SKELETONS or the ATTACKER) turn. Anyone striking a burning skeleton with an unarmed strike or natural attack takes 1d6 points of fire damage."
SO IF IT"S THE SKELETON"S TURN:
SO, if a skeleton doesn't have anyone next to them at the start of their turn doesn't take damage, BUT if the skeleton immediately moves next to someone at the beginning of their turn to attack, do they then take the D6 of fire damage from being adjacent?
The fire aura only does damage at the beginning of the turn. So if there is no one next to the skeleton, no one takes damage from this ability. Then it moves to the wizard and starts giving him hugs, no one takes damage from this ability. When the wizard starts his turn though, he will take 1d6 fire damage for starting next to the skeleton (assuming the fighter didn't murder the skeleton).
I guess what's meant is that if you start your turn near a burning skeleton you get dealt damage.
So if the skeleton moves to stand beside you and your turn comes up you get burned.
Burning Skeleton - Fiery Aura (Ex): "Creatures adjacent to a burning skeleton take 1d6 points of fire damage at the start of their turn."
The word "their" refers to the creatures, not the skeleton. (Because "Creatures" is plural, but "skeleton" is singular)
Any creature that begins its turn adjacent to a burning skeleton takes 1d6 points of fire damage.