1) This elegant version: Besieged in court, routed in eight states, accused of favoring blacks and Latinos at the expense of Asians and whites, affirmative action — a major legacy of the civil rights era — is once again the subject of uncomfortable scrutiny.
2) Comes from this:
Affirmative action (a major legacy of the civil rights era) [which has been] besieged in court, routed in eight states, accused of favoring blacks and Latinos at the expense of Asians and whites is once again the subject of uncomfortable scrutiny.
You can pre-position those three verbal phrases that are predicates in the clause. There are THREE verbal phrases introduced by three different verbs: besieged, routed and accused.
A simpler example:
The boy who was attacked by lions is now at home with his parents.
Attacked by lions, the boy is now at home with his parents.
Another:
The lady who was charmed by her co-workers, entranced with her boss and filled with hope, drove home at high speed.
Charmed by her co-workers, entranced by her boss and filled with hope, the lady drove home at high speed.
You can shift the predicates to the head of the sentences to avoid using heavy relative clauses.