What’s the advantage of using this language over Rust, Go, Java (or any other JVM language), or Python?
Elixir (really, the Erlang VM it's built on) is designed for highly concurrent, reliable systems. For my org, building a distributed webapp, we chose it over the languages you listed because of how much you get "out of the box" with Elixir.
Saša Jurić's talk "The Soul of Erlang and Elixir" (https://www.youtube.com/watch?v=JvBT4XBdoUE) is a good introduction to why someone might choose Elixir. The payoff is the slide (https://imgur.com/gallery/G6lfUW6) where he compares the stack of a "traditional" web app (Nginx + Ruby on Rails + Go + Redis + Mongo + Cron + Upstart) to the Erlang rewrite of the same app, which is... just Erlang.
This isn't to say that Erlang ships a complete replacement for Redis, Mongo, Upstart, etc.—but it provides most of the functionality that most apps will need, such that your dependency tree can be much simpler. It's really nice to not have to become an expert in all the infrastructure surrounding a normal web app and instead just learn Erlang/Elixir.
I hadn't seen that slide, it's pretty compelling. Is there a story for typechecking or a typed language that runs on BEAM? This is one aspect that holds me back a little.
The Norm library (https://github.com/keathley/norm) is an attempt at higher-level, more expressive type checking which I think shows a lot of promise, but I've not adopted it in a real project yet.