What happens to a losing player's cards in Exploding Kittens?
They go back into the Discard Pile.  


  EXPLODING KITTEN
  ... Unless you have a Defuse Card, you're dead.
  Discard all of your cards, including the Exploding Kitten.


Since there are one more Exploding Kitten Card to start with than normal, there still can be a winner.  


  STREAKING KITTENS
  ... 6. Insert enough Exploding Kittens back into the deck so that the total number of Exploding Kittens is equal to the number of players - this is 1 more Exploding Kitten than normal because the Streaking Kitten Card allows a player to hold an Exploding Kitten in their hand without exploding!


Exploding Kittens Rules
Exploding Kittens with Streaking Kitten Rules