Elixirs are automatically applied when purchased (even while dead) if there are already 6 items in the inventory.
Elixirs applied in this fashion while dead will still last for the entire listed duration once the player has respawned.
Only one Elixir effect may be active at a time. Activating a new elixir will remove the old effect.
It's a great tool for initiators to make sure their team can follow up.
Elixirs are often bought towards the end of long games when builds are all completed and the extra gold continues to accumulate. They can also be bought mid game to give a temporary stat advantage during ganks and teamfights.
Path of Iron has a similar effect to the removed Captain enchantment.