Your bridge crew are like any other item. You equip them FROM your inventory, then go to the assign tab and assign them. It's not clear why this isn't made more transparent, but then again it's not too stressful to figure out. I suspect it's this way so you can trade them?
While some people are just not looking in the right place for acquired BOs, the first two BOs I was awarded in mission never appeared in my officer pool or inventory. There's definitely a bug regarding receiving BOs from quests.
If you get a Bridge Officer from a Quest there is an item in your inventory. Just a Equip it and the BO is assigned to you!
Id you get a BO from a NPC just go to you assignments by pressing U and click on assignments. Then there is a Unkown BO on the right side of the UI above it its stated "Bridge Officer Candidates" just right click on it and assign or send him to hell.