How does God reconcile us to himself?
God's justifying grace works by faith to bring reconciliation to God through the atoning sacrifice of Jesus Christ.
Catechism for the Global Methodist Church, question 56
4Surely he has borne our griefs and carried our sorrows; yet we esteemed him stricken, smitten by God, and afflicted. 5But he was pierced for our transgressions; he was crushed for our iniquities; upon him was the chastisement that brought us peace, and with his wounds we are healed. 6All we like sheep have gone astray; we have turned every one to his own way; and the LORD has laid on him the iniquity of us all. 12Therefore I will divide him a portion with the many, and he shall divide the spoil with the strong, because he poured out his soul to death and was numbered with the transgressors; yet he bore the sin of many, and makes intercession for the transgressors.
16For God so loved the world that he gave the unique son, in order that everyone who trusts in him should not perish but have eternal life.
10For if while we were enemies we were reconciled to God through the death of his Son, much more, having been reconciled, we will be saved by his life. 11Not only that, but we also boast in God through our Lord Jesus Christ, through whom we have now received the reconciliation.
18All things are out from God, who reconciled us to himself through Christ and gave us the service of the reconciliation. 19That is, God was in Christ reconciling a world to himself, not counting their trespasses to them, and placing in us the word of the reconciliation.
13But now in Christ Jesus you who once were far off have been brought near by the blood of Christ. 14For he himself is our peace, who made both one and broke down the middle wall of the fence, the hostility, in his flesh, 15having abolished the law of commandments in decrees, so that he might create the two in himself into one new human, making peace, 16and might reconcile both in one body to God through the cross, having killed the hostility in it.
19because in him all the fullness was pleased to dwell 20and through him to reconcile all things to himself, having made peace through the blood of his cross, through him, whether things on the earth or things in the heavens
17Hence he had to be made like his brothers in every respect, so that he might become a merciful and faithful high priest in matters pertaining to God, to make propitiation for the sins of the people.

Question 56 reaches the center of the Wesleyan way of salvation. The answer paraphrases ¶102.5 of the Book of Doctrines and Discipline: "God's justifying grace is received by faith to reconcile us to God through the atoning sacrifice of Jesus Christ, what God does for us." Article IX of the Confession of Faith states the doctrine formally: "penitent sinners are justified or accounted righteous before God only by faith in our Lord Jesus Christ," never "through our works or merit." Article VIII adds the ground of it: "God was in Christ reconciling the world to himself." The catechism places it after preventing and convincing grace because pardon comes to the penitent.
The answer has three parts, and the references support each. First, reconciliation is needed because there was enmity. Paul says "while we were enemies we were reconciled to God through the death of his Son" (Romans 5:10), and those who "once were far off have been brought near by the blood of Christ" (Ephesians 2:13). Second, reconciliation is accomplished by an atoning sacrifice. Isaiah foresaw it: "he was pierced for our transgressions; he was crushed for our iniquities; upon him was the chastisement that brought us peace, and with his wounds we are healed" (Isaiah 53:5). The writer to the Hebrews explains why the Son took on flesh: "so that he might become a merciful and faithful high priest in matters pertaining to God, to make propitiation for the sins of the people" (Hebrews 2:17). Third, reconciliation is received by faith: "everyone who trusts in him should not perish but have eternal life" (John 3:16).
The initiative throughout is God's. "God was in Christ reconciling a world to himself, not counting their trespasses to them" (2 Corinthians 5:19). ¶102.5 also says that justification "ordinarily results in the direct assurance of 'God's Spirit witnessing with our spirit that we are children of God,'" the assurance Wesley preached. Paul describes it: (Romans 5:11).
A Methodist rests here. Pardon is a finished gift received by trust, and it cannot be improved by effort or lost by a bad day; it can only be received or refused. The believer who has received it has also been given, in Paul's words, "the service of the reconciliation" (2 Corinthians 5:18): the work of carrying the same peace to others.