The effects of a single species coming extinct can be very devastating for the food chain that that species happened to be in. i.e. perhaps it was the single (or most important) food for some carnivore or perhaps it was the only animal that ate some specific plant. In the former case the carnivore is likely to get into trouble too and in the latter case the plant might no longer have a natural enemy and become a pest.
In both situations you can get a chain reaction that could get out of control. Note that of course not animals are the only species that are required to get a food chain running but the problem is that we might not know what the implications are. Potentially the loss of a single species *could* cause a chain reaction causing many other species to be lost.