It was common knowledge, though not for most of those (with some exceptions like legacy gaming fans) coming into the PC world much after the early 2000s.

Terminate but Stay Resident was the original DOS function call to allow a utility program to exit but stay in memory waiting for interrupt signals (it will have hooked into those interrupts before calling TSR) to respond to. Utilities that operated in this manner (mouse drivers, and many other useful things) became collectively known as TSR programs and then just TSRs.

There were later other methods of leaving code resident, implemented by various memory managers for background tools that needed to use RAM outside the first 1MB (which in reality meant outside the first 640KB because of how the PC memory map was arranged), but the name TSR stuck to cover them generally.