Is static electricity really so dangerous?
Yes.
The conductive paths inside an IC are really small, so it doesn't take much energy through them to vaporize them.¹
There are millions of such paths inside the ICs of an Arduino, and it only takes damage to one of them to break the device. It is possible that you could get lucky and break some feature that you aren't using, but this is not a good gamble to take.
Can I really destroy something simple like an Arduino module if I touch it?
What makes you think an Arduino is "simple?" Microcontrollers are among the most complex and delicate objects humans make. Fabergé eggs are simple and durable by comparison.
How does this happen?
It happens the same way you get a discharge when touching a doorknob. Whenever the path through the device is a better path for the electricity than leaking away through your shoes and the air, it will take that path.
Generally, the device is a good path because it is plugged into a power source, which means there's a ground path, which is a low-impedance path to a much lower voltage potential.
Some of an Arduino's input pins will be protected against static discharge, but probably not all. Even those that are protected can be killed with enough hits. Protection doesn't make a pin invulnerable, it just allows it to withstand a certain amount of ESD energy. Like any armor, hit it enough times with enough energy, and you can break through.
This is not to say that unplugging the device is a good solution to the problem, however. For one thing, it defeats much of the protection built into the device, because some of it works by shunting the dangerous energies to ground. When you unplug the device, you remove that path, so now the electricity is forced to take a different path through the device, one without this protection. An unplugged device still has other conductive paths that lead to a lower voltage potential. Consider that humans and shoes are poor conductors, yet we manage to get static buildups, and can be electrocuted.
Anything more I should know?
Go to a good electronics tool shop and look at all the antistatic products available. They're made for a good reason. Buy some. Use them. :)
Footnotes:
The sort of static discharges you feel when touching a doorknob range between 5 kV and 15 kV depending on several factors;² the peak current can be 1 A. The static electricity is equalized in about a microsecond and doesn't pass through your heart, so it is merely annoying to a human but potentially fatal to a semiconductor gate.
The human body can also charge to less than 5 kV before discharging. There is a range where there is still enough energy in the discharge to damage a sensitive IC, but where the energy is too low for your nerves to sense it.
Dry air allows the human body to store more charge because it leaks away slower than in moist air, so that the energy in a static discharge generally gets higher in winter. Your shoe type also affects the amount of energy you can deliver in a static discharge, as does the carpet. Then you have effects like how moist your skin is, whether your fingers are callused, etc.
