The formal meaning of the word is important to understanding how it is being used in memes and by young people in the context you describe, so let's cover that first:
The meaning of "triggered" in a psychological context is that something has "set off" or "sparked" an emotional response in someone. For example, someone who has been trapped in an elevator may develop a fear of confined spaces, and finding themselves in a small space may "trigger" that fear.
It comes from the idea of cause and effect - pulling the trigger on a gun causes it to fire. In fact there is also the expression in British English "to light the touch paper", which uses the analogy of lighting the fuse on an explosive to describe something that causes anger or excitement in somebody.
The memes you refer to feature various people portrayed as having some kind of militant ideology along with an inferiority complex; that is to suggest that they believe everybody is against them and rant about their ideology at the merest mention of it. The idea behind the meme is to suggest that it doesn't take very much to "trigger" their complex and start ranting.
If someone use the word this way about themselves, my instinct would be that is was hyperbole, and with a knowledge of this meme. Similar expressions have become popular in recent times - one that springs to mind is "I'm cured" to mean you are tired of, or have had enough of something.
So it doesn't specifically mean someone that is upset, although if somebody's anxiety or complex was "triggered" they might be upset. Depending on how it was used (and I think its usage will be limited to the demographic of people who are really into quoting internet memes) it could equally mean someone is angry, scared, traumatised etc.
If people are asking "why so triggered?" as you say I would interpret this to mean "why has this made you angry/upset/whatever?" rather than simply "why are you angry/upset/whatever?".