They originated in the feminist blogosphere.

In the field of traumatic stress studies, a trigger is a stimulus that provokes the recollection of a traumatic memory, accompanied by intense distress in a person with Post-traumatic Stress Disorder (PTSD). Today, the term signifies a stimulus that provokes intense emotional distress in people in general, including those without PTSD. 

The primary rationale for trigger warnings is the belief that “forewarned is forearmed.” That is, proponents of trigger warnings believe that if people are told that they are about to encounter a distressing stimulus, they will be able to mobilize their coping skills and experience less distress. However, experimental research shows that the opposite typically occurs; people randomized to conditions where they receive no trigger warnings typically experience less distress than the people randomly assigned to receive trigger warnings prior to reading or seeing potentially distressing content. Trigger warnings tend to increase anxiety, but the magnitude of the anxiety response is modest and it does not persist. These findings support the conclusion that trigger warnings do not work as intended.

There is some evidence suggesting that exposure to trigger warnings increases how “central” a trauma is to one’s sense of self. Unfortunately, the heightened centrality of trauma in one’s life is associated with subsequent increases in the severity of one’s Post-traumatic Stress Disorder symptoms.

I do not know. However, the research suggests that it would either increase anxiety or have no effect at all.

Both traumatized and non-traumatized groups typically experience more anxiety about content preceded by a trigger warning relative to when they encounter the same content without a trigger warning.

Experiments suggest that people rarely opt out from viewing or reading content preceded by a trigger warning. However, it is possible that those most prone to opt out never volunteer to participate in such experiments in the first place.

Clinical psychology and counseling students who require trigger warnings are probably choosing the wrong profession! Indeed, the stories they will encounter while treating clients can be very intense and will not be preceded by warnings.

What I have always done is explain the syllabus, noting the topics covered and so on without “warning” them that the content may be distressing. As noted above, there is no evidence that trigger warnings “work” in terms of reducing distress. Some students may request trigger warnings as signifying their sensitivity and respect for their identity. Of course, not “warning” them signifies even greater respect for them as competent adults.

Based on what we know about anxiety disorders, I would guess that exposure to challenging (or anxiety-provoking) situations where they perform well despite (initially) experiencing anxiety will likely bolster their self-efficacy and perhaps their resilience against subsequent stressors.