What happens when I successfully attack a player's home stronghold?
The token, representing the garrison of the stronghold, is removed once it has been taken by an enemy player. 

If the original owner of that territory retakes it, the token is not replaced.

Relevant quote from the rule book, page 26


  If a Garrison is defeated in combat (whether defending by itself or
  with other friendly units), it is permanently removed from the game,
  regardless of the number of Sword or Fortification icons on House
  cards played in the combat.