Specifically means particularly, or regarding one option out of several. You specifically requested that your birthday cake be chocolate with raspberry icing, so you were quite disappointed with the vanilla cake you were served instead.
The adverb specifically comes from the Latin specificus, and shares a root with special and species. If you specifically choose something, you have made a special choice about it. If you single something out specifically, it's particularly significant. A documentary about this history of childhood might have been a real eye-opener, specifically the part about how kids used to go to work at 12 or younger.