You've certainly used them, but been unaware of it.
Made a menu selection? State Machine.
Interacted with a dialogue box/an HTML form? State Machine.
Used a dropdown/popup suggestions list? State Machine.
Written async code? State Machine.
Used a VM? State Machine.
Used a compiled executable? State Machine.
Any time you don't have all the input available when you start (equivalently, can be interrupted by some other task) either you explicitly, or a tool on your behalf implicitly, will need to reach for a state machine. Even if all the input is in principle available, if it is too large for practical processing, piece-by-piece processing entails a state machine (however rudimentary).
https://lamport.azurewebsites.net/pubs/state-machine.pdf
"safety" (nothing bad ever happens) and "liveness" (something good eventually happens) are two key properties for systems, and they both have easy formalisations in state machine terms.