Supports are probably the first or second most important members on a team once you hit a certain level of play, IMO (with junglers as the other 1 or 2 spot).

They act as force multipliers for your team and as force dampeners on the other team. If your ADC is good, a good support will net him double kill after double kill and rapidly snowball the team. If your ADC is bad and the other team is very good, a support can shove off the jungle ganks (nami wave, leona ult, morgana binding) and keep the ADC up till he reaches tower.

Very often a team will ward if it sees theres already good vision in place; that is, if the support is reliably warding and sweeping, they will too (it jobs their memory, perhaps). If on the other hand a support does not ward well, the rest of the team cannot safely ward, will get ganked a lot, and will rapidly end up huddling in lane with no vision as their board control disappears.

Supports also play pivotal roles during teamfights; take soraka, whose heals over a 6 second period add another full healthbar to your vital ADC while interrupting dangerous spells, or a nami who can split their team and break up a wombo combo; or leona who can make that crucial pick on the adc.

Other roles can do similar things too-- assassins for example make those crucial picks-- but the support is doing so many other things that allow the core team to actually do the work of killing. In the presence of a good team, the support turns a 5v5 into a 6v5 or 7v5 in terms of power.

The big downside of support is that if your team is lackluster enough, acting as a force multiplier does very little. If you are in bronze 5 trying to rise in MMR, you probably want a role where your own skill can brute-force the team to victory; getting an ADC kills who is just going to immediately die in a teamfight doesnt really help you.