They're to prevent condensation from pooling on the table itself. This can warp or otherwise mess up a wooden surface. On other materials it can just leave ugly rings that you have to clean off.

Condensation. if wooden tables get exposed to it over time it ruins the wood/leaves stains

They are to prevent condensation from damaging the surface beneath them.

In addition to the condensation, I have also used coasters for under hot drinks like coffee as they can discolor or even deform some surfaces.

If all you drink is room temp liquids you don't need coasters.

You have to respect the wood.

It depends on the humidity of where you're at. On a warm day in Houston, for example, it's not at all uncommon for water to condense on the outside glass of a cold drink, run down, and eventually damage the wooden table it's sitting on. A coaster will help prevent that damage.

Additionally to what else was said, I've seen them used in beergardens to cover the drink to prevent bugs from drowning themselves in your drink. I guess then the coasters don't serve to protect the table anymore.

Every mother always insists in using coasters and makes sure to explain to you a millions times why is that and will burn the house down if they find a cup mark.

Ever have your coaster stick to your glass or cup. Follow this trick! Shake some salt on the coaster. It will never stick!

Nah, they're so your wine bottles can coast on tables.

Seriously, that's what they started as.

Using them to protect tables came later

Try thirsty stone coasters. I found a really large thirsty stone trivet which I use. Combine with tervis tumbler for the ultimate drinking setup. Lol. For spilling; try a straw. Or try a baby sippy cup?