Justin Taylor has an interesting and helpful post that takes the reader step by step through the second half of Romans 1. In an addendum to his post, he quotes a sermon by John Piper addressing the question of why Paul connects homosexuality with idolatry in this chapter.
Piper's explanation is an important aspect of the Christian view of homosexuality that often gets overlooked. Christians frequently refer to the idea that there is a created purpose for sexuality and for our bodies, and that homosexuality is wrong because it goes against this purpose, but I think Piper's description (from Romans 1) of homosexuality as a dramatization of the rejection of our proper relationship to God, in direct opposition to heterosexual marriage's illustration of Christ and the church, is helpful in giving us a deeper understanding of the issue.
The reason Paul focuses on homosexuality in these verses is because it is the most vivid dramatization in life of the profoundest connection between the disordering of heart-worship and the disordering of our sexual lives. I'll try to say it simply, though it is weighty beyond words.
We learn from Paul in Ephesians 5:31-32 that, from the beginning, manhood and womanhood existed to represent or dramatize God's relation to his people and then Christ's relation to his bride, the church. In this drama, the man represents God or Christ and is to love his wife as Christ loved the church. The woman represents God's people or the church. And sexual union in the covenant of marriage represents pure, undefiled, intense heart-worship. That is, God means for the beauty of worship to be dramatized in the right ordering of our sexual lives.
But instead, we have exchanged the glory of God for images, especially of ourselves. The beauty of heart-worship has been destroyed. Therefore, in judgment, God decrees that this disordering of our relation to him be dramatized in the disordering of our sexual relations with each other. And since the right ordering of our relationship to God in heart-worship was dramatized by heterosexual union in the covenant of marriage, the disordering of our relationship to God is dramatized by the breakdown of that heterosexual union.
Homosexuality is the most vivid form of that breakdown. God and man in covenant worship are represented by male and female in covenant sexual union. Therefore, when man turns from God to images of himself, God hands us over to what we have chosen and dramatizes it by male and female turning to images of themselves for sexual union, namely their own sex. Homosexuality is the judgment of God dramatizing the exchange of the glory of God for images of ourselves. (See the parallel uses of "exchange" in verses 25 and 26.)
It's important to put all of this in the context of Taylor's full post on Romans 1, a chapter he refers to as "a universal text that applies to all of us apart from Christ—what we are, what we do, and what we would do apart from God’s restraining and redeeming grace."