What is the point of TF's red card?

It deals in an AoE damage & slows(duh), plus if you're in need of CC & damage, red is better than yellow since red card deals more damage than yellow card.

It does aoe damage, so using it in a teamfight or to hit multiple cs is when it's best thrown

It does tons of damage, use it when you do not need to stun stuff or when you do not need mana. It is a awesome tool for wave clear.

only for csing if you have enough mana to not need to use blue

in practically every combat situation gold card is better

What it's best for is teamfights when everyone is stacked (and you don't need to stun right away) or when laning, you can throw it at a minion near your laning opponent and hit them with the splash damage. Same with a turret if someone is low, just toss a red card at it, and kaboom.

AoE-dmg? AoE-Slow?

When you are chasing and you wana kill both of them instead of 1 ;)

AoE damage when youre in bigger fights. More damage than yellow card for when you dont necessarily need the stun for the kill.