It is often used towards the end of the game to help with a final push or to help defend. It is best for a champion that relies heavily on physical damage, but it has benefit for all champions due to the additional health.
The health bonus can be used somewhat like Heal, instantly granting you extra life that could save you from death. This is a common tactic to turn the tide in an early fight or to survive a gank. For example, having one ready to use is an effective way to prevent a death early or mid game to RequiemKarthus.
Because the extra health is not gained by health regeneration or healing, it can be considered an effective counter to Ignite.
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.
v3.6Elixir of Fortitude is meant to be an "all-in" burst of power for players looking to take control of a situation. With its low cost, however, players were simply defaulting to it as a general starting item with no significant repercussion. Increasing the gold cost of Elixir of Fortitude aligns it better with the risky strategy it was intended for.