The term Terminate and Stay Resident made sense when it was originaly used. A basic overview was that for one program to provide extra functionality to the user after another program was started under a single task, real mode OS what it would do is load all the information and code it needed into memory, then tell the OS that it was quiting but not cleaning up it's memory. The OS would oblige and the program would 'stay resident' in memory after it had 'terminate'd. This program could say be a sound or mouse driver. Another program would come along and it would know about that some driver that it might like to use could be resident. It would ask the OS for a list of resident programs and their addresses in memory, and seeing one it was interested in it would take it's memory address and knowing that it alway provided a certain function at a predefined location after it's start of memory it would make a call to that location and the TSR could then take an action on the running programs behalf.
DOS could also be considered resonable as many OSes for micro-computers of the day were stored in ROM not on a disk, thus the distinction that the OS was stored on writable media was important to some people.