Why does Paul refer to Abraham so often?

Paul refers to Abraham frequently because Abraham was saved by faith plus nothing—before law, sacrifice, or circumcision—so he serves as the pattern for all who enter salvation by faith alone. Paul uses him to show that children of faith are Abraham's children without Gentiles becoming Jews.

Les teaches that Paul alludes to Abraham probably more than any other Old Testament character, and in Romans 4:1 Paul comes back to Abraham. In Galatians, Paul says those who are children of faith are also children of Abraham. This does not mean a Christian becomes a Jew; a Gentile remains a Gentile, and salvation does not make a Jew out of anyone. Abraham was saved by faith plus nothing. He did not sacrifice, did not practice circumcision, and had no law; he simply did what God said to do. Therefore, just as Abraham entered in without law, sacrifice, or circumcision, we too enter in by faith alone. That is why Paul uses Abraham over and over.

Romans 4:1

What shall we say then that Abraham our father, as pertaining to the flesh, hath found?

How do believers enter into salvation according to Paul's example of Abraham?

Believers enter salvation by faith alone, just as Abraham was saved by faith plus nothing. Abraham had no law, sacrifice, or circumcision; he simply did what God said. Paul uses Abraham to show that Gentiles are not made Jews but enter in the same way—through faith apart from those works.