to lead someone out: to conduct another person or creature from inside a place to outside it
This is a plain word for one party taking another out of somewhere. It joins the ordinary verb for leading or bringing (ἄγω) to a prefix meaning "out," so the picture is simple: someone is inside, and someone else conducts them outside. Jesus leads the disciples out as far as Bethany, soldiers lead him out to be crucified, a shepherd leads his flock out of the pen. The word always has a leader and a led; it is never just a person walking out on their own.
In Acts the word keeps turning up for prisoners:
- an angel opens the prison doors at night and brings the apostles out
- the Lord brings Peter out of Herod's prison while the guards sleep
- the magistrates at Philippi are made to come and bring Paul and Silas out in person
It is also the standard word for the exodus, used again and again for God bringing Israel out of Egypt. Those two settings are connected. The Greek translation of the Old Testament used this verb for the exodus so steadily that it became the sound of rescue, so when Luke has an angel bring the apostles out of jail, he is telling a small exodus story in the old words.
The verb itself, though, is not a religious term. It stays the everyday word for taking someone out, and what makes any one instance an act of deliverance is who is doing the leading and what the person is being led out of. The same word carries a condemned man to his execution.