What is the meaning behind the phrase “PALMAM QUI MERUIT FERAT”?

Answer 1

"Let he/she who earned it receive the first prize."

Ferat is the third person singular present active subjunctive of the verb fero, ferre, tuli, latus which means to carry or bear. In this context, it's being used as a polite command.

A more literal translation might be something like, "Let he/she receive (ferat) the first prize (palmam, accusative direct object) who earned it (relative clause)."