Here "believes" means that one puts his trust in Christ. AT: "it is through the gospel that God powerfully saves those who put their trust in Christ" (See: [[rc://en/ta/man/translate/figs-explicit]])
Paul speaks about the gospel message as if it were an object that God could physically show to people. You can translate this in an active form. AT: "God has told us that it is by faith from beginning to end that people become righteous" (See: [[rc://en/ta/man/translate/figs-activepassive]])
Here "righteous" refers to those who trust in God. AT: "It is people who trust in God that he considers right with him, and they will live forever" (See: [[rc://en/ta/man/translate/figs-explicit]])