If I have a card like Treasure Keeper, which has an effect that says I 'may cast [a] card without paying its mana cost' (the card being selected by Keeper's ability), what happens if the card selected has an additional cost that must be paid for the spell to resolve, such as Bone Splinters? Will Splinters fizzle because the only thing I 'may' do is play a card for free, which in this case will mean I am unable to pay the additional cost to make the card resolve?
Treasure Keeper's ability dispenses you from paying the spell's mana cost. A card's mana cost is the mana symbols printed on its top corner. Any other cost can/must still be paid.
When you cast Bone Splinters with Treasure Keeper's ability, you still need to sacrifice a creature. And if you don't have any to sacrifice, you just don't cast Bone Splinters at all - it doesn't "fizzle", it goes to the bottom of your library because it was revealed but not cast.
Additional costs are part of casting the spell, not of resolving it.
When you cast Bone Splinters with Treasure Keeper's ability, you still need to sacrifice a creature. And if you don't have any to sacrifice, you just don't cast Bone Splinters at all - it doesn't "fizzle", it goes to the bottom of your library because it was revealed but not cast.
Additional costs are part of casting the spell, not of resolving it.