"One of those listening was a woman named Lydia, a dealer in purple cloth from the city of Thyatira, who was a worshiper of God. The Lord opened her heart to respond to Paul's message." NIV translation
This too can be a somewhat puzzling verse to read as we are told that Lydia was a "worshiper of God" but then we are told that "the Lord opened her heart". You see, many people want to worship God but that does not change the fact that, to have a right relationship with God, we must accept his Son, Jesus, as the payment for our sins.
"When she and the members of her household were baptized, she invited us to her home. 'If you consider me a believer in the Lord,' she said, 'come and stay at my house.' And she persuaded us." NIV translation
This verse affirms a pattern that occurs throughout the entire Bible. That is the pattern of people being baptized when they come to Christ. When Lydia accepted the message of Jesus Christ, we see that her entire household was baptized. Likewise, we see the pattern of hospitality in that, as a Christian, she offered her home to Paul and his companions.