As I recall it bei8ng explained to me, way, way back -- the suffix "ster" means "female of the profession*. A spinster, as Marcus notes, was this a female spinner -- that is, one who spins fibers to make yarn.
Married women with girl children had their daughters do the spinning -- it was something that one could learn pretty quickly, and all you needed was a drop spindle (spinning wheels were a luxury). And spinning the yarn was felt to be the dull, boring part -- you really wanted to get on to the weaving on the loom.
A spinster was this seen to be a woman who didn't have anyone to do the spinning for her, and was reducede to doing it herself, and often did it for pay, selling the spun yarn to others. Thus, an older woman without children, hence the idea of a Spinster as an older, unmarried woman.
* just don't tell the Teamsters that. Evidently, through time, some of these suffixes changed significance.