If half the rabbits in a given population are killed off, there will be impacts both up and down the food chain.
Looking down the food chain, the things that the rabbits eat will not be eaten as much. Therefore, their population will increase. They may crowd out other plants that compete for their niche (if the rabbits didn't eat those too).
Other organisms that eat the same things as rabbits will also flourish. They will have less competition for the food.
Up the food chain, animals that depend on the rabbits for food will see their populations drop. This is because they will not have as much food to eat.