Microbiology is the study of unicellular or cell clusters, whether they be eukaryotes (fungi and protists), or prokaryotes (bacteria). Although argued that they may not be alive, viruses are also studied in detail by microbiologists. In theory, a big part of microbiology is the study of the pathogens that cause death to people. For instance, studying what caused influenza would fall under either virology (since influenza is a virus) or could fall under the job of a microbiologist who would be well rounded in virology as well as the microbiology.
However, that does not answer the question at hand. What reason is there to study microbiology? To answer that, first allow me to explain a little more about what is studied in the field of microbiology.
Eukaryotes were mentioned and those are cells that have a nucleus. Fungi and protists were mentioned as examples of eukaryotes. It is fungi that are responsible for athletes foot and ringworm, but also cause much more harmful diseases to people that are immuno-deficiencies. Cryptosporidium is a protist that can cause intestinal problems, cause diarrhea, and dysentery. That can cause death.
Moving on to the prokaryotes, these are what I have always found to be most interesting. These are the bacteria that you hear about so often. In the news, MRSA probably popped up. Methicillin-resistant Staphylococcus aureus is the actual name for MRSA, but this is basically a really bad infection that cannot be treated. While studying the bacteria, it can be understood what causes these horrible infections we have.
And finally, viruses were mentioned. Viruses are responsible for influenza and chicken pox and then as horrible as HIV/AIDS. Viruses are an interesting class because they are arguably not alive and yet they are called alive. However, the study of virology goes into much more detail about these bizarre, yet fascinating...things.
The point that I hope was taken from the above three paragraphs is that the study of microbiology is the study of very small things that cause so much death. The Black Plague was caused by something that, if studied by microbiologists, could be prevented. Unfortunately for them, in the 14th century, there were no microbiologists or even an understanding of what caused it.
We study microbiology so that we can try and find cures for illnesses. We study microbiology so that, hopefully, epidemics like the Black Plague never happen again. We study microbiology because microbes (eukaryotes, prokaryotes, viruses, etc) can do so much damage even when they are so small. The bigger question would be: why NOT study microbiology? It truly is fascinating.