This same thing happens in real life on college campuses. If a combat veteran attending college on a GI Bill is in a gen-ed history class, and there's video of, say, World War II combat, that could easily trigger that veteran's PTSD. Similarly, people who have survived rape may also be suffering from PTSD, and descriptions of sexual assault can trigger a traumatic episode. A trigger warning is a simple note in the syllabus saying, "Hey, we're going to discuss something that may cause some of you to relive a traumatic life experience. Please prepare accordingly." Those last three words are important, because that combat veteran or that rape survivor will likely actually prepare accordingly. It's a pretty complex idea, I know. Some people just can't wrap their head around it.
What it doesn't mean -- but what most people think it means -- is, "We might mention something that will hurt your feelings. Go hide in this special room so the bad words don't hurt your precious, fragile ears."
"Take your legitimate medical issues and get out!"
As for safe spaces, I'll just describe what they were where I went to school. I graduated from the University of Notre Dame, famous for (among other things) being one of the least LGBT-friendly universities in the country. A number of Notre Dame professors display a small rainbow sign outside their offices that simply read, "This is a safe space." It was a sign that told students who were struggling with their sexual identity (on a campus that is, again, not friendly to gay people) that they could talk to that professor without fear. Maybe a minority student could come to that professor when he didn't know how to handle the guys in his hall casually dropping the n-word around him and needed help valuing his cultural identity. That's it. Safe spaces aren't some club where nasty liberals sit around and bash cis straight white people. That's what coffee houses and drama clubs are for. In my experience, they're basically just small environments where students could go to not casually have "n****r" or "f****t" slung in their direction.
"Come on in, we're not horrible pieces of human garbage here."