It can be used for just about anything you can use a microcontroller for, but it seems particularly popular for art or music-related projects. For instance, audio synthesis, controlling large numbers of programmable LED strips, custom MIDI controllers, and so on.
I've used mostly the Teensy LC, 3.2 and 3.6 boards. I use them as kind of a cross between a GPIO and a data acquisition board. For instance it's easy to plug one into USB and write a program that lets you talk to it, e.g., from Python, to read voltages, turn things on / off, etc.
From there, it's not a huge quantum leap to program some lengthy sequences of operations into the Teensy itself rather than in Python, e.g., averaging 1000 consecutive analog readings, or interfacing with some interesting chip via SPI bus. Migrating this functionality onto the Teensy reduces the amount of USB traffic needed to implement a function, so the "system" as a whole becomes more efficient.
Also, the Teensy can do stuff in real time based on its internal clock, that can be difficult on a modern operating system.
The T3.6 has built-in floating point, which makes some kind of programming easier.
So far I haven't created any stand-alone uses to speak of. I'm mainly interested in stuff that extends the functionality of my regular PC. I've even created such mundane things as a USB temperature sensor. While not the cheapest or most elegant way to do it, I was able to spin it up in no time flat for something that needed it.