The weighing actually has many meanings but the most important thing it represents is spiritual cleanliness and the lightness to traverse dimensions along with the daily duty this involves to achieve. An extremely dirty and trapped soul resides in undesirable dimensions and hence is 'judged' to weigh too much to traverse. Like much of the imagery of Lord Anubis, Lady Ma'at, Lady Nephthys and Isis [Lady Astarte], this also deals with Astral Projection and the Astral body coming out of the Middle Chakra.
The feather is a metaphor for the serpent energy hitting the 6th and Crown Chakras unobstructed, which takes us to 'seventh heaven' and makes us light as a feather, so to speak.
At the back of the bow in Cupid and Eros imagery there is a feather, which also has a lot to do with this.
The heart is also equated with the body itself in Egyptian texts, so this also relates to the body itself and its physical health, to an extent.
Complicating matters, in contrast to the physical body meaning, the heart also represents the aspect of the soul that forms the core personality regardless of one's body or current heredity, along with the ties to others formed in a lifetime. As far as Anubis Himself is concerned, it also represents the courage and the undying loyalty aspect of Lord Anubis character in this light, which in this case has to be interpreted as bravery and courage in wanting to live continuously and be oneself, to live life, the reasons for wanting to come back [which may involve romance, bravery, beauty, hard work, building skills and other things that make life worth living] which is not always easy.
This deals with the refinement of the heart. By living, one can inscribe superior experiences and model oneself into a better, more complex, more radiant and 'lighter' person. One can also go the other way though - and then the heart, which is just heavy rotten meat, is thrown to Ammit or Sobek. Although it seems contradictory at first glance, the body and soul were thought to be in tandem in this sense. Clearly if you're an obese blob weighing 600lb, it's pretty hard to refine anything...
Beyond justice and measurement, the scales represent union too - matter to aether, spirit to reality, and so forth. In many languages the word for Scale is related to the word yoke in English, or yoga in Sansrkit. Libra also deals with these themes.
Jewish archaeologists, perverts and idiots deciphered the meaning of this into being some sort of shit test of a melting yenta reading off your moral transgressions, like mikhael weighing souls or shit peter at the gates.