If A sees B (Black) and C (Black) he does not know which he is wearing until he realises,
B (Black) must be seeing C (Black) and A
C (Black) must be seeing B (Black) and A
If A's Hat is white then both B & C will be stuck in an insolvable problem as they cannot know for a fact if they are wearing Black or White (both being possible).
As B & C stay silent so long and A realises the puzzle must have an answer he concludes he must be wearing a Black Hat.