Extra Kevlar allows you to give your teammates additional armor protection that lasts until they die. You cannot administer this buff to yourself, or to a Soldier who already has the Kevlar ability.
This allows an Engineer to decrease the damage that their teammates take by a small amount (20%). It is administered like a Weapon Buff, by throwing them the upgrade. As such, the Engineer needs to be close enough to the target in order to reach them.
Although stated otherwise in the in-game description, Engineers can buff Soldiers who have the Kevlar Vest ability. The effect is only half as effective though; adding 10% resistance to the Kevlar Vest's 20%.