Jamieson, Fausset and Brown on 1 Timothy 3:7

7. a good report--Greek, "testimony." So Paul was influenced by the good report given of Timothy to choose him as his companion (Ac 16:2).

of them which are without--from the as yet unconverted Gentiles around (1Co 5:12; Col 4:5; 1Th 4:12), that they may be the more readily won to the Gospel (1Pe 2:12), and that the name of Christ may be glorified. Not even the former life of a bishop should be open to reproach [Bengel].

reproach and the snare of the devil--reproach of men (1Ti 5:14) proving the occasion of his falling into the snare of the devil (1Ti 6:9; Mt 22:15; 2Ti 2:26). The reproach continually surrounding him for former sins might lead him into the snare of becoming as bad as his reputation. Despair of recovering reputation might, in a weak moment, lead some into recklessness of living (Jr 18:12). The reason why only moral qualities of a general kind are specified is, he presupposes in candidates for a bishopric the special gifts of the Spirit (1Ti 4:14) and true faith, which he desires to be evidenced outwardly; also he requires qualifications in a bishop not so indispensable in others.