en_tn/act/13/43.md

When the synagogue meeting ended

Possible meanings are 1) this restates "As Paul and Barnabas left" in verse 42 or 2) Paul and Barnabas left the meeting before it ended and this occurs later.

proselytes

These were non-Jewish people who converted to Judaism.

who were speaking to them and persuading them

"and Paul and Barnabas were speaking to those people and persuading them"

to continue in the grace of God

It is implied that they believed Paul's message that Jesus was the Messiah. Alternate translation: "to continue to trust that God kindly forgives people's sins because of what Jesus did" (See: rc://en/ta/man/jit/figs-explicit)