And by why I mean what was the inspiration behind it?
This is an extremely simple story in some ways. Spencer Silver, a chemist at 3M, developed an unusual adhesive in 1968. On one hand, it was not very strong: but on the other hand, it didn't leave a residue and stayed tacky, even when pulled off of something. Another chemist at 3M , Art Fry, was annoyed by how bookmarks fell out of his hymnals, and when he learned about Silver's adhesive, he thought it would be a good thing to use for that. They made some up and started using them around their offices. From there it made its way into other offices of 3M, where it was found useful, and it was offered for sale in 1978. Here's the patent, if you're curious how the stuff works.
So, in one way a rather normal happy industrial tale: bright researcher creates new thing, another bright researcher discovers application for new thing, new thing goes out and makes money, clever researchers are famous. The interesting part of it is that the adhesive was evaluated by male executives, who for years thought it was not that important because it was not strong, and couldn't be used on a tape. Newer strong adhesives, like epoxies, cyanoacrylates, phenolics, were very important to applications that needed a lot of strength- building a boat, laminating wood beams from veneers, bedding structural fasteners in concrete. 3M also made tape: masking tape, duck tape, etc. This stuff didn't fit either application. When the sticky note pads were put out in the offices, it was the female office staff who seized on them and began using them in the way you find now- posting memos, etc. And because it wasn't on tape, or strong, it wasn't considered worth male executive interest and so it was a woman, Pat Gaudio Edwards, who was given the project to market. She, like the female office staff, realized its possibilities. It's hard to avoid conjecturing that, had women been involved in the development process earlier there might have been Post-it notes in 1969 instead of 1978.