Why Teemo support?

The biggest reason is Utility.
I know that a lot of people would disagree with that, but that doesn't change the fact. He is the only support character that can give 100% map vision to his team. If you play your cards right, and if you have a proper team that doesn't flame you since the champ select; it is near impossible to lose a teamfight in the jungle, even if the entire enemy team gets sweeping lens. They can't clear all your mushrooms, and for each lens they get; they lose more map vision. Long story short: you will always have more vision than the enemy team if you play right, but how the hell do you play it right?

Teemo is fragile and fairly easy to kill with a little bit of CC.
Remember that most of the time your team will be toxic, because you are trolling.
Sadly a lot of people are toxic no matter what you do, and you can see their true colors in the champ select thanks to Teemo. Keep a cool head, that is the first step. Rely on hit and run; don't engage all in and the most important part is: Place your mushrooms properly.

Mushroom types

Tactical

Corners and narrow pathways are the key locations, you can observe these by randomly clicking on somewhere on the map, and seeing how your character moves. It will be walking through corners around the jungle, and those locations are the most important ones. Make sure you put a mushroom in every single bush around you, and make sure you prepare the dragon. Teemo's biggest impact is the control on the dragon. Make sure you don't let the enemy can't move freely, make sure you communicate with your adc and your jungler. If they co-operate, it will be almost impossible to miss dragons.

Combat

Teemo isn't that strong in the lane, so try to pull the enemies to the jungle. After level 11, you can start throwing mushrooms around in teamfights, but remember that you are still the support and create safezones for your adc. Teemo mushrooms will make it really easy for your adc to kite and the enemies will be less likely to chase your allies if you create withdrawal points. Some people might panic or they might be blind, make sure you keep pinging them the safe zones if they start running away randomly.