en_obs-tn/content/47/02.md

opened Lydia's heart

That is, "enabled Lydia."

she and her family were baptized

This can also be translated as, "they baptized Lydia and her family."

so they stayed with her and her family

This was the normal custom of the day for people to provide hospitality to visitors in their homes. There were no immoral motives in this arrangement.