I thought Azura's Star would fill with souls when I killed something (bear, deer, etc.) with my bow of soul snatching. But I capture the souls and it shows still being empty?

Is there something I need to do to activate it or something?

Do you have other unfilled soul gems? i believe the game attempts to choose an appropriate one and Azura's star is considered one of the higher graded soul gems. (which are used last when lower grade souls are absorbed if others are present)