How to obtain elixir

You obtain elixir  in the game mode Show Down.  In Show Down, boxes of elixir are distributed in the map.  You collect the elixir by breaking the box and walking over the icon.  Maps are different but elixir box locations for a single map remains the same throughout all games played.  You can also get elixir by killing other players in Show Down.  If they have already collected some elixir, they have a chance to drop around two to three elixir instead of only one.

What to do with elixir

You do not really have a say in what to spend the elixir as once you pick it up, your brawler gets stronger.  They get more health and do more damage.  Because of this, the more elixir you collect, the easier it will be to win the game.  All elixir collected will reset once the game ends.