An abuse vector is any feature of a website that can be used to harm someone. This can take the form of threats to a person's safety, communicating with someone who does not want to be communicated with, revealing of personal information, and many others.
A humble text field could be an abuse vector, depending on where the text is going and who's able to see it!
There have been many high-profile cases of targeted online harassment against public figures, but it can happen to anyone, and has happened to an overwhelming number of people. According to a Pew Research survey conducted in 2017, roughly 4 in 10 Americans have personally experienced online harassment. While more men have experienced harassment than women (44% of men and 37% of women), roughly half (53%) of young women ages 18 to 29 say that someone has sent them explicit images they did not ask for.
Harassment takes a huge toll on its victims, with survey respondents reporting mental and emotional stress, financial loss, difficulty finding a job, and more as a result of being harassed.
It also makes sense from a business perspective to care about online harassment. 13% of adults who witnessed online harassment decided to stop using an online service to protect themselves. Users may not stick around to spend money, generate valuable data, or click on ads if they don't feel safe.
You can read more statistics from this survey at http://www.pewinternet.org/2017/07/11/online-harassment-2017/.