I know the support in lane are used as sustain for the ad carry, but what role do the offer in the later game in team fights?
Usually the support stays with the AD Carry at bottom lane.
They are there because a lot of reasons
They buy more wards than their teammates. So they can ward more of the map (How do I ward properly in League of Legends?), also they usually buy Oracles so they can hunt down enemy wards (Jungles also usually buy oracles, because for a Jungler it's easier to walk around looking for enemy wards).
At this point your function is to keep your group alive as much as you can. Make the opponent lose their focus and stop attacking your allies (knocking up, silence, stun, etc). Keep Warding and if possible keep an Oracle activated.
To be a good support, you need to:
It really depends on the support your playing. For example soraka's main job stay alive, silence, and heal. Taric- stun, stay alive, heal with the high cd heal. And Janna who has a shield, knock up, dis-engagement, and slow. Most of the time as a support you should be getting aura items like wota, shurelyas, randuins, starks, etc. These items are only active when you stay alive so you should stay alive as much as possible. Really supports excel in early game and mid game. What role do they offer in late game in team fights? Like I said it depends on the support your playing, all supports bring something different into team fights. But all in all they bring; survive ability, aura's that give more armor, mr, AD, attack speed, and Ap. There are also active items that slow down the attack speed and speed of enemies and speed your own team up. They bring crowd control; slows, knock ups, stuns, silences, etc. All in all all support have different skills so they bring different things to team fights.
The general rule of support characters is to keep your allies alive. And this is done, well, by supporting them.
you don't want your carry to die. First because it give other team advantage in gold plus free farm while the carry is dead, and second because you generally will have difficulties to defend alone against two (you are too fragile, or you can be ignored) Your next role is to allow your carry to farm as much as possible. You can do that by zoning the enemies out, healing shielding, ... To stay alive you also need to ward accordingly to the situation. If you can also offer some kill to your carry thanks to your crowd control, or by baiting one champion, this is good too.
you will be able to roam more. This is ideal time for warding. It's also the beginning of team fights. For those fights, you need to learn/know who's gonna be the focus of the enemy team and try to protect it. This is not a general rule, sometimes it's better to keep your abilities for other champions (for example, champion with stealth can generally take care of themselves, a good Tryndamere will pop his ultimate and escape). Most of the times, you will have a tank in your team. He should not be your priority target. He normally has good built-in sustainability and even if he will get the focus in the first seconds of the fight, a good team will prioritise one of the carry. You want to protect your damage dealer at all cost, cause they will win the game for you. Remember, you have to save allies at your life cost if necessary, but you should do it reasonably. Do not save someone that's going to die no matter what.
If you are a support champion with crowd control, use them to disrupt enemy attack. Janna can scatter enemies in multiples groups allowing your team to put their grasp on one isolated champion, thus giving a 1 kill advantage to the team.
Let's talk summoner spell and stuff.
First heal and CV
If your champion can heal with skill, you would generally drop heal spell. I took this spell when I was unsure about the champion I was playing. Now, I never take it. This can, however be a good spell. It gives more sustainability in lane and can save in some situation. But in my opinion, it's not worth a good crowd control spell or a flash. The best use you will have of the heal, is to take enemy by surprise when they think they can kill you. You can use this usually early or mid game. After that, you need your partners to be able to use the small time you give them to kill other champions or you will die anyway.
Now the CV. I never ever go out without it. Why this? because you always need it. Even if you don't know your jungler or your team is bad. CV helps you save lifes, control the map, spy, control crowd and kill. It saves your life when you can check location before going their. If you spot 5 champions in ambush you win. You can then control the map. By knowing where the enemies are, you can ambush them, you can defend key points (baron, dragon). It helps spying by getting a view on champions that are warding, allowing you to get some oracle or pink ward to get that out. It helps sometime to control crowd. When revealed by CV, its not uncommon for enemies to scatter. If you can take advantage of this, free kills for your team. And you can also kill with CV, how so? What are CV useful for: negating fog of war and bush stealth. I once killed a enemy champion because he was going back home teleporting in bushes. Hopefully there was a minion wave here. CV helped my negate the stealth of the bushes, allowing the minions to see the enemy porting back, and directly attack him. Interrupting him I then just had to catch him on my way back from the river.
Well this is difficult. Usually, teamates want you to get auras. Good teamates will ask you to get the auras items they don't get themselves. So first, look what your team don't have. Try to avoid getting Will of the Ancient if someone already has it. Second think. What will helps the team more? MR, attack speed reduction, spell vamp?
My advice, your ultimate as a support can make or break a teamfight.Use it wisely. You also need to be able to pay attention to not get targeted easily and save your summoners spells to disable the one targeting ur ad carry or allow him to do more damage.
They generally provide support through their abilities such as Shurelya's Reverie and Randuin's Omen. They also should support other characters if they have shields to help tank towers or stuns to help start team fights.