A large number of adjuvant chemotherapy drugs are also DNA damaging agents. When the drug is transported across the cell barrier, it interacts either directly with DNA (cisplatin, 5-FU), or it interferes with cellular machinery for maintenance of DNA quality (camptothecin, topotecan). A few others such as taxol are even more indirect, instead hitting the "structural" parts of the cell which hold it up or together. But in general, the chemotherapy drugs damage DNA in some way.
If you read what is adjuvant chemotherapy in this blog, you will find that these drugs hurt all cells. Understanding that the way they work is by damaging DNA, also makes it obvious why these drugs are hurting all cells. But importantly, because cancer cells are dividing rapidly, they must continually make DNA so are susceptible to the DNA damaging agents.
Most interesting are the class of drugs for adjuvant chemotherapy known as the topoisomerase inhibitors. Topoisomerase is a cellular machinery that winds and unwinds DNA as it's about to be replicated. DNA exists as a double helix, but must be "opened up" so that it can replicate and divide into two for each daughter cell. The topoisomerase inhibitors are drugs which interfere with the topoisomerase preventing it from winding or unwinding the DNA effectively. As a result, the DNA of the cells become damaged due to incorrect unwinding and winding.
There is also an very interesting compound known as "cisplatin", which contains the metal platinum. It is a very "simple" drug in the sense that it contains an atom of platinum, surrounded by just a two nitrogens and two chlorines. The other compounds mentioned have much more complicated chemical formulas and structures. Their complex structures give them the ability to break DNA by mimicking it, or by specifically shutting down proteins and enzymes that maintain DNA integrity. But the cisplatin is slightly different. The platinum atom has a structure which allows it to draw in atoms from other molecules. In particular, when placed near a DNA molecule, the platinum atom draws in DNA atoms and bends it. Breaks in the DNA is thought to occur when it is bent by platinum. The cisplatin family has spawned a series of other platinum compounds such as oxaliplatin with various additives that enhance its cancer killing ability. However, for some reason certain patients become resistant to platinum therapy. While nothing can prevent cisplatin from binding to DNA, the cancer cells likely evolve DNA repair strategies or short-circuit the cell-death processes which generally lead to destruction of cancer cells.
The resistance of cisplatin adjuvant chemotherapy can be potentially overcome with other types of drugs such as taxol.