Read 1 Timothy 5 at Bible Gateway.
The majority of this chapter forms a chiastic structure:
1 Tim 5:1-19
1a) 1 Tim 5:1a, Do not rebuke an older man;
1b) 1 Tim 5:1b-2, Exhort older/ younger men as fathers/ brothers, older/ younger women as mothers/ sisters;
1c) 1 Tim 5:3-10, Care of real widows, family to provide for their own widows (chiasm);
1d) 1 Tim 5:11-12, Refuse younger widows as they desire to remarry + cast off their first faith;
central axis) 1 Tim 5:13, “And besides they learn to be idle, wandering about from house to house, and not only idle but also gossips and busybodies, saying things which they ought not;”
2d) 1 Tim 5:14-15, Charge younger widows to marry + some already turned aside;
2c) 1 Tim 5:16, Let the believers relieve their own widows that the church may relieve the real widows;
2b) 1 Tim 5:17-18, Let the elders who rule well be counted worthy of double honor;
2a) 1 Tim 5:19, Do not receive an accusation against an elder except from two or three witnesses.
1 Tim 5:3-10 (1c pair above)
1a) 1 Tim 5:3, Honor widows who are really widows;
1b) 1 Tim 5:4, But widows with children, let them repay their parents; as is good + acceptable;
1c) 1 Tim 5:5, She is really a widow who is left alone, trusting in God + continuing in prayers;
central axis) 1 Tim 5:6, But she who lives in pleasure is dead while she lives;
2c) 1 Tim 5:7, These things command, that they may be blameless;
2b) 1 Tim 5:8, But anyone not providing for his household has denied the faith + worse than unbeliever;
2a) 1 Tim 5:9-10, Real widows: 60+, been wife of one man, known for good works.
1 Timothy 5:25
These chiasms are about the proper care of widows, the proper treatment of the elderly and elders, and the proper responsibilities of family members for their own households. The central axis’ of both concern the pitfalls waiting for younger women who do not have any responsibilities to husband, children, or household. This was written in a time when women did not typically work outside the home as providers; the men of the family having that responsibility. The warning is not about the younger men being idle, since presumably the men worked to provide for themselves and their families.
The old proverb is “Idle hands are the devil’s playground.” Humans need something worthwhile to occupy their time and attention. Children fill this need by school and play, women by managing their households, and men by working to provide for their families. Someone with character will find or create a worthwhile occupation for themselves; someone without character, being idle, wanders about from house to house (i.e., sticking their nose in everybody else’s business), being busybodies, gossiping, and saying what they should not.
Speaking of saying what they should not, do not rebuke the elderly, but encourage them, treating them with all honor. Do not gossip about or bring accusations against the elders or leaders who are over you, as dishonor and disrespect is not befitting the household of God.