First of all, electromagnetic radiation is a whole range of stuff. Starting at the low end (frequency-wise) you’ve got long waves (useful for submarines to communicate through polar ice but not much else), radio waves, microwaves (which are really just higher frequency radio waves), infrared light, visible light, ultraviolet light, x-rays, gamma rays, and cosmic rays.
The lower frequency stuff is less harmful than the higher frequency stuff. This is because part way through the ultraviolet part of the spectrum, electromagnetic radiation becomes “ionizing” meaning that it can strip the electrons off of atoms and create ions. Ionizing radiation is known to be very harmful to all kinds of stuff. This is why ultraviolet light causes things to fade if it is left out in the sun, and it’s why sunlight causes skin cancer and why x-rays cause cancer. So some of the ultraviolet stuff and everything else higher (x-rays, gamma rays, cosmic rays) is ionizing and is very dangerous. This is the stuff that most folks are talking about when they use the term “radiation”.
Lower frequency stuff isn’t so dangerous. You can shine a flashlight on your skin and it won’t hurt you (you wouldn’t want to do the same thing with a flashlight that put out the same relative level of x-rays). Cell phones use microwaves, and despite millions of panicked people on the internet saying otherwise, these are too low in frequency to be harmful in the same way that x-rays and gamma rays are.
That doesn’t mean that the lower frequency stuff is completely harmless though. We are used to visible light being harmless because most of the time when we encounter it, the amount of it striking us is harmless. A flashlight, for example, is too low in power level to be harmful. Get the power level high enough though and you can cause all kinds of damage. This is how microwave ovens cook things. Cell phone levels, like a flashlight, are completely harmless. Microwave ovens use the exact same type of radiation (microwaves) and will cook food. Visible light can cook things, too. Just ask anyone who has ever fried an ant with a magnifying glass and sunlight. Lasers can certainly cause quite a bit of heat.
Your eye’s retina is very delicate, and can be much more easily damaged by both ionizing and non-ionizing radiation than other parts of your body. Sunlight will burn your retinas to the point of destruction but will only make the skin on your arms feel warm. Your skin is just tougher, and your retinas are just that easily damaged. So it’s not surprising that the very bright light from a welder can damage your retinas.
Welders also put out UV light. If you are looking away from the welder you can prevent the visible light from damaging your retina, but the UV light might still strike your corneas. Remember, UV light is ionizing. It is more dangerous. In this case, repeated exposure to your corneas will likely cause cataracts to develop. For similar reasons, a lot of sunglasses these days have lenses that wrap around to the side to prevent the sun’s UV light from striking your corneas from the side.
So basically, welders can hurt your eyes in two ways. First, the intense power levels of visible light can burn your delicate retinas in the same way that a microwave oven cooks food, just by having a large amount of electromagnetic radiation there. Second, the UV light can cause cataracts, even if it doesn’t strike your retina. Welding goggles prevent both problems.