Since we have been justified by faith (like Abraham), we experience peace with God rather than wrath (5:1). The wrath of God has been satisfied in the death of Jesus so that those who are in Christ by faith experience peace, not wrath. Paul uses an aorist passive participle (Δικαιωθέντες) to indicate we did not justify ourselves, but also that this justification is an accomplished fact (Kruse, Romans, 225).
Our experience of peace, however, is a present tense verb (ἔχομεν), having been justified in the past, we are now in a state of peace with God. I should mention the famous textual variant here, some manuscripts read ἔχωμεν, a subjunctive verb rather than indicative. This alternate reading is supported by both Sinaiticus and Alexandrinus, but in both cases a later hand corrected the text to an indicative. In short, Paul appears to be making a statement using the indicative rather than making an encouraging statement using the subjunctive.
The peace Paul has in mind is not inner peace, but rather a cessation of the enmity humans have with God. In Romans 1-3, humans were enemies of God, but now they can be in a state of peace with God. Ephesians 2:11-22 has a similar idea. After he describes Gentile alienation from God, he declares it is the work of Jesus on the cross that “brings close” Jews and Gentiles. This is the idea of reconciliation: Gentiles who were apart from Israel, and the Jews who were apart from the Gentiles, are now made into something new.
Thiselton points out reconciliation was not used in the Jewish writings of the Second Temple period, nor is it found in the Old Testament. He considers this an example of Paul’s genius, using a word for familiar to Gentile readers in order to get make the Gospel clear in terms they would understand (Discovering Romans, 124).
Since we are in a state of peace with God, we have now access to the Father (5:2a). In order to have access to a king, one must have appropriate status. The word translated access (προσαγωγή) is used by Xenophon, for example, to describe those who have access to the Persian king Cyrus (Cyr. 7, 5, 45). The same word appears in Ephesians 2:18 to describe Jews and Gentiles having access to God the Father through the same Spirit.
The one who is in Christ has the appropriate status to enter into the presence of God through the Holy Spirit, later Paul will expand this metaphor by describing us as adopted into the family of God, so that we can call God abba, father. This is in contrast to anyone who tries to obtain salvation through works. Since they are not justified by faith (and adopted into the family of God), they never really do have access to God.
In Second Temple period Judaism, one did not directly approach God. Only the high priest could enter the presence of God in the Holy of Holies, others can only approach so far (court of men, women, gentiles, etc.) In the worship of Greco-Roman gods, one did not approach them directly nor were humans granted access to a god. This access to the Father is a remarkable claim in the ancient world!