How does the Holy Spirit lead us to repentance?
He convinces the world of sin, of righteousness, and of judgment.
Catechism for the Global Methodist Church, question 22
8But as for me, I am filled with strength, with the Spirit of the LORD, and with justice and might, to declare to Jacob his transgression and to Israel his sin.
7But I tell you the truth: it is profitable for you that I go away. For if I do not go away, the Advocate will not come to you; but if I go, I will send him to you. 8And when he comes, he will convict the world concerning sin and concerning righteousness and concerning judgment. 9Concerning sin, because they do not believe in me. 10Concerning righteousness, then, because I am going to the Father and you will see me no longer. 11Concerning judgment, then, because the ruler of this world has been judged.

The first thing the Holy Spirit does in any sinner is bring that person to repentance. Article III of the Confession of Faith says it in the words the catechism uses: "He convinces the world of sin, of righteousness and of judgment." The article is quoting Jesus in the upper room, so the church teaches here what the Lord himself said about the Spirit's work.
Jesus told the disciples that his departure would be to their profit, "for if I do not go away, the counsellor will not come to you; but if I go, I will send him to you" (John 16:7). Then he described the counsellor's first task: "when he comes, he will convict the world concerning sin and concerning righteousness and concerning judgment" (John 16:8). Jesus explained each term. Sin, "because they do not believe in me" (John 16:9): the root sin is unbelief toward Christ. Righteousness, "because I am going to the father" (John 16:10): the Father's welcome of the risen Son vindicates him and exposes every other standard of righteousness. Judgment, "because the ruler of this world has been judged" (John 16:11): the cross has already passed sentence on the powers of evil. The same Spirit was at work centuries earlier, filling Micah "with strength, with the spirit of the LORD, and with justice and might, to declare to Jacob his transgression and to Israel his sin" (Micah 3:8).
Methodists call this work convincing grace. ΒΆ102 of the Book of Doctrines and Discipline says convincing grace "leads us to what the Bible terms 'repentance,' awakening in us a desire to 'flee from the wrath to come.'" The same paragraph records that Wesley counted repentance among the three main doctrines of Methodism, with faith and holiness, and called it "the porch of religion." Conviction is grace because it comes before any merit: God is already at work in the sinner who feels convicted.
A Methodist can welcome conviction. When a sermon, a passage of scripture, or a prick of conscience shows a person their sin, the Spirit is at work, and the right response is honest confession and a turning toward Christ. The Spirit who does this uncomfortable work is the God confessed in the previous question, and his conviction is the love of God drawing a sinner home.