What did Paul mean by saying believers are children of Abraham?
Paul meant that believers are Abraham's children by faith, not by physical descent or by becoming Jews. Abraham was saved by faith plus nothing, without law, sacrifice, or circumcision. Likewise, Gentiles enter salvation by faith alone, so all who believe share Abraham's spiritual lineage.
Les explains that Paul comes back to Abraham in Romans 4, and in Galatians he says that we who are children of faith are also the children of Abraham. This does not mean that when you become a Christian you become a Jew; a Gentile is a Gentile, and salvation does not make a Jew out of anyone. The connection is that Abraham was saved by faith plus nothing. Abraham did not sacrifice, did not practice circumcision, and did not have law; he simply did what God said to do. So even as Abraham was saved without benefit of law, sacrifice, circumcision, or any of those things, we too enter in by faith alone. That is why Paul uses Abraham over and over.