The rat (쥐, jwi) opens the Korean zodiac (십이지신) as the first of the twelve animals — according to the traditional story, it won its position through cleverness rather than size or strength, a detail that colors how the rat is read in dreams generally: small, quick-witted, resourceful.
In an agrarian context, rats hoarding grain gave the animal a real, if backhanded, association with stored wealth — seeing a rat with food, or multiple rats gathered together, is traditionally read as a small but real financial gain, close in spirit to a money dream but on a more modest scale.
The same animal carries a pest's reputation too, and that ambivalence shows up directly in the dream's darker variant: being bitten by a rat, or rats damaging your belongings, is read as petty loss or minor betrayal — something small being nibbled away rather than a dramatic setback.
A white rat specifically is read more favorably across most interpretations, and a rat entering the house follows the same general "something small arriving" logic that runs through most animal-entering-the-house dreams on this site — good news, just modestly scaled.