First of all, we must acknowledge that not everyone thinks that it is imperative to protect endangered species. There are many people who believe that there is no great harm in allowing various species of plants and animals to become extinct.
That said, there are also many people who do believe that preservation of endangered species is important. Their arguments can be both ethical and pragmatic. On the ethical side of the argument, some people argue that we do not have the right to destroy species. From this point of view, other species have the right to exist and we act unethically when we take actions that prevent them from existing.
The pragmatic side of the argument has two main aspects. One of these aspects holds that each extinction brings us close to some sort of ecological disaster that we may not be able to foresee. We do not know everything there is to know about the biosphere and how its various parts interact. As we remove pieces (species and even whole ecosystems) we arguably are weakening the system to the point where it may collapse at some point with unforeseen consequences. The other aspect of this argument is a bit more tangible. This argument is that many plant and animal species end up being useful for us in terms of medicine. When we destroy species, we are destroying our chance of ever finding uses for those species. In other words, a species that we drive to extinction might have been the species that would have helped us find a cure for cancer or Alzheimer’s.
For these reasons, some argue, it is imperative to protect endangered species.