to come to someone and take a close look, meaning to act on what is found: to visit and help, to look after, to pick out
Underneath this word is a look that leads to action. Its parts mean roughly "to look upon," and ordinary Greek used it for an inspection: a commander reviewing his troops, a doctor examining a patient, an official checking on how something stands. What separates it from simply looking at something is that the one who looks is expected to do something about what he sees.
So when the New Testament says God "visited" his people, it does not mean he dropped by. It means he came and took their situation in hand. Zechariah blesses God for visiting and redeeming Israel in one breath, and the crowd at Nain cries that God has visited his people just after a dead man is raised. The word arrived in Christian use already loaded, because the Greek Old Testament regularly used it for the Hebrew verb "to attend to" (פָּקַד), the standard word for God stepping in, whether to rescue or to call to account.
The human uses run along the same line, and each of them keeps the working edge of the word:
- going to the sick and the imprisoned, where the visit is itself the help
- looking after orphans and widows in their trouble, which James calls religion at its purest
- going back to people you already know to find out how they are getting on
- looking a group over in order to pick the right people out of it
That last use, when the apostles tell the church to look out seven men, shows how wide the range runs. It is the same attentive looking, only aimed at choosing rather than at relieving. The noun built on it (ἐπισκοπή) turns the act into a role, and that is how the early church came to speak of overseers.