I looked through 3 separate web pages to figure it out (it was relatively easy), and thought it'd help to gather it into a single page of 10 steps.
What's Elixir? Elixir is a programming language that superficially looks a little like Ruby, but more importantly is designed to be dynamic, functional, and capable of building concurrent, distributed, low-latency and fault-tolerant systems (perfect for the web!).
It does that by running on the Erlang VM (like a Java VM, not like a VirtualBox VM), and that also gives developers access to Erlang’s ecosystem (this reminds me of how Clojure runs on the JVM, providing access to Java's ecosystem).
What's Phoenix? Phoenix is an Elixir framework implementing the server side of a web framework for modern web applications. Like Ruby on Rails or Python's Django, but Phoenix is written in Elixir and derives benefits from that (in addition to whatever other advantages it may have as a framework - we're not comparing frameworks today).
(1) Install Homebrew
To begin, you need Homebrew installed on your MacOS. Homebrew is a package management system that makes installing lots of developer software much easier on Macs.
Once installed, especially if it's been a while since you've last used Homebrew, make sure to update it (commands after "$" are to be typed in your terminal window):
$ brew update
Do a self diagnostic to make sure it's all okay, carefully read the instructions it provides:
$ brew doctor
(2) Install Elixir
Elixir requires Erlang to be installed. Fortunately, the Homebrew package will automatically make sure of that.