The judgment of his peers means the trial by a jury. It is a term of expression borrowed from Magna Charta.
In common law, judgment of his peers is a trial of twelve men. [Webster v. Reid, 52 U.S. 437 (U.S. 1851)].
Following is an example of state constitution guaranteeing a trial by the judgment of his peers. “No member of this state shall be disfranchised, or deprived of any of the rights or privileges secured to any citizen thereof, unless by the law of the land, or the judgment of his or her peers, except that the legislature may provide that there shall be no primary election held to nominate candidates for public office or to elect persons to party positions for any political party or parties in any unit of representation of the state from which such candidates or persons are nominated or elected whenever there is no contest or contests for such nominations or election as may be prescribed by general law." [NY CLS Const Art I, § 1].