One of the most popular and compelling real world uses of Lua is for embedded scripting. It's easy to embed, it's easy to use, for a scripting language it's blazing fast.
I'm sure Lua has other sweet spots, but that's the one I'm most familiar with.
I've mostly experienced it through World of Warcraft -- it's embedded there as the language used to write the game's user interface[1]. Then it's exposed to third-party authors as an API, so they can write addons for the game.
It's also embedded in Wikipedia[2]. Complex templates can be written with Lua modules, rather than relying on wikitext.
I was introduced to Lua via writing a Wireshark plugin for packet verification. It was hard in the beginning but dove right into working with a C backend. I'd say that's an approachable side project if you want to explore Lua.
I'm sure Lua has other sweet spots, but that's the one I'm most familiar with.