How does one enter the US after making application for a green card?
When you apply for permanent residency from abroad, at the end you receive a US immigrant visa. At that point, you are not yet a permanent resident and cannot get a green card. When you enter the US with that immigrant visa, you immediately become a US permanent resident, and that immigrant visa, stamped upon entry, serves as your temporary proof of permanent residency while you wait for your green card to arrive in the mail.