Watsonian explanations/interpretations are premised on how the world is built and the personalities are described, as if we were describing a world that existed separately from the author. Doylist explanations/interpretations are grounded on the intent and needs of the writer in crafting their story and how decisions are made to set up plot, theme, etc.
To give an example, if we’re asking why Varys was kept alive after the Sack of King’s Landing, a Watsonian explanation is that Robert Baratheon and Jon Arryn were generally pro-amnesty, Varys was a foreigner and thus didn’t have long-term ties to the Targaryens in the way a Westerosi nobleman would have, and he was good at a dirty job that a lot of nobles wouldn’t necessarily want to hold. A Doylist explanation is that GRRM needed Varys to be alive and to have been associated with the Targaryens for the plot of ASOIAF to work, so he wrote it that way.