To me, "cutleaf" is descriptive of its leaves. "Wort" is an Old English word meaning "plant" so that part of the name is also obvious. "Tooth" is not so obvious. I have never dug up a toothwort but all the sources I have looked at assure me that there is a toothlike appearance to its segmented rhizome. My two oldest wildflower fieldguides identify it as "Dentaria laciniata". This is its former scientific name. The old Genus, "Dentaria" refers to those toothlike segments on the rhizome.
Paracelsus, a Swiss physician and alchemist who lived from 1491 to 1541, developed a philosophy called "The Doctrine of Signatures". Its premise was that plant shapes and colors were signs that these plants would cure diseases and problems suggested by the appearance of the plants. Thus, because the rhizome of the toothwort resembles teeth, it is useful for curing problems of the teeth. Jakob Bohme, a German mystic who lived from 1575 to 1624, suggested that God marked objects such as plants with a "signature" to suggest their purposes for man. The Doctrine of Signatures continued to be a popular belief for centuries, especially among herbalista.
There is no scientific evidence that the uses of plants are determined by their appearances.