I understand that words like "hell" and "damn" are cursing the person they are directed to, but why would words such as "shit" and "bitch" be viewed as curses?
Cursing as an offense comes from throwing a curse or an evil spell; words, when carried with intention can affect people on emotional levels, like a curse. The secondary sense is swear, a closely related word "use bad language" (early 15c.) developed from the notion of "invoke sacred names." Thus you make use of a bad language, in the original sense, not because of coarseness, but determined by the misuse of it. Later it became the word of choice to the bad election of language in the form of cursing.