You know, I still haven't heard any part of that story other than the punchline.
1. Constitution mandates that the King must give royal consent to anything passed by the Riksdag and the Realm Council.
2. King refuses to sign certain things passed by the Riksdag and the Realm Council that he doesn't like. He tries to veto it.
3. Confusion. What do we do now? We can't enforce it as law if the King refuses to give royal consent!
4. The Riksdag comes up with ingenious legal fiction. Even though the person of the king may not consent to the bill, the office of the king must, by definition, consent to it.
5. Therefore, the actual signing of the bill is just a formality. If the king cannot perform that action himself, then a stamp can be used just as well.
6. Riksdag and Realm Council establish that this is how the Constitution is to be interpreted, and orders artisans to create stamp of King's signature.
7. The King of Sweden is LITERALLY a rubber stamp.