Step 1: Identifying the context.
In the story "The Rattrap" by Selma Lagerlöf, Charley's Granddad receives an envelope with a stamp. The stamp has a picture of the King of Sweden.
Step 2: Analyzing the details.
The stamp's picture of the King symbolizes Sweden's royal presence. It is a subtle indication of how official matters were marked by the country's royal identity.
Step 3: Conclusion.
Thus, the picture on the stamp of the envelope was that of the King of Sweden.
Final Answer: \[ \boxed{\text{The picture on the stamp of the envelope was that of the King of Sweden.}} \]