Why are there so many negative effects of chemotherapy?

Cell damage: Most "traditional" or cyto(cell)toxic(damage) chemotherapy damages cancer cells and normal cells. Cancer cells are preferentially damaged because they are growing abnormally. Killing rapidly dividing/growing normal cells causes many of the potential side effects of chemo including hair loss, diarrhea, mucositis, etc.

Non specific: Unfortunately, standard chemotherapy does not differentiate between normal cells and cancer cells. That is why some normal cells are affected like cancer cells are. These are typically cells that divide quickly like hair follicles, bone marrow, the lining of the GI tract. That's where you see toxicity.