The reaction works with claim, pillage, or any card ability that discards cards from an opponent’s hand or deck (such as Without His Beard or Wardens of the West). It also works with abilities that require cards to be discarded from hand as part of their cost (for instance, Consuming Flames).
Characters discard cards using pillage one at a time, so you will normally only be able to draw 1 card. However, it is possible for a single character to discard multiple cards using pillage (with Scouting Vessel, for instance), in which case you can draw more.
If Hotho Humpback is used during a challenge, players check reserve in turn, but all the cards one player discards for reserve are discarded simultaneously.