From another dictionary (Dictionnaire étymologique de la langue française, Bloch & von Wartburg):

essai lat. exagium “pesée” (dér. de exigere, au sens de “peser”), qui est attesté dès le IVe s. au sens de “essai”

To weigh has been used with a figurative meaning (to measure, to test) for a long time, surely because it was one of the few accurate measurements that you had in Antiquity. For example Latin pensare became both French words peser and penser.