Choice
Legal language
"Elected" is voting/choosing language, but also theological—Calvinist doctrine of election meant God chose who would be saved. Dickinson turns it around: she's doing the electing.
Death's moment
"Sense from spirit files away" is the exact moment of death when body and soul separate. "Subterfuge"—all the social pretenses and deceptions—ends when you die.
Courtroom reveal
"Figures show their royal front" means people reveal their true selves, like witnesses finally telling the truth. The "mists" of earthly confusion clear away at death.
The chosen one
After three stanzas of "when death comes," she finally reveals: "Behold"—look at—this one soul I chose above all others. "Atom" makes the beloved both tiny and fundamental, like atomic theory.