It's like a sunburn. The radiation damages the DNA of the cells. DNA damage like that triggers a cellular self-destruct mechanism. It's a cancer-prevention mechanism. So over the next few days as the skin or corneal cells start to go through their normal replication cycle, the DNA damage is discovered and causes those cells to die off. This causes an injury just like if another mechanism like heat or chemical damage, had killed those cells. Inflammation, oozing, pain, scarring.
A couple of years ago, I had a mishap with a moderately powerful laser, meant for engraving, grazing quickly across my left eye (I was wearing appropriate eye protection).
It wasn't until the next day that it started feeling like someone poured sand in it. It took a few days to return to normal.
Appropriate eye protection does not provide 100% protection. Outside of a welder's helmet, maybe, even the best eye protection won't stop a reasonably powerful laser beam directed straight into your eye. It will reduce its intensity, though. The main benefit of wearing it is that even the diffuse, scattered light bouncing off of the walls is hazardous, and the eye protection stops that. Without it, you couldn't safely be in the same room as an operating laser even if the beam never comes near you.
If I had not been wearing it, there is an excellent chance that I would have suffered permanent vision damage. The eye protection I wore did exactly what it was supposed to do.
Unfortunately no, like a sunburn sometimes you don't feel it until later. This can also happen when hiking across snow covered plains under clear very bright skies. It's called snow blindness.
Its like a sunburn blister. Takes a while to develop, usually whilst you're asleep. When you wake up, you open your eyes and rip the blisters open. I'm told it feels like hot sand being poured in your eyes at that point. Definitely one to avoid!
That's the big danger with solar eclipses. It's not bright enough to hurt at the time. The damage only starts to become apparent after a few hours or the next day.
When I went to see a total eclipse I was surprised that even a tiny slit of the sun was still hard to look at. Not sure why anyone would try to stare at that.
galileo (the galileo) used to stare at the sun for extended periods of time (without a telescope), to the point he could identify sunspots. But IIUC he looked at it near the horizon when it's attenuated by more atmosphere.
As a kid (~6 years old), I once tried to look at the sun for as long as possible, competing against myself. Luckily, I didn't do any noticeable damage, but I feel like someone should have told me about UV radiation and the damage it can cause to the retina.