re: What does it mean that God gave them over to their sins?

In these verses (vs. 24-27), Paul explains what God's judgment looks like for people and societies who reject Him. Just a few verses earlier in vs. 18, Paul states that "The wrath of God is revealed (present tense) from heaven against all ungodliness and unrighteousness of men who suppress the truth in unrighteousness." All mankind has rejected God, even though there is sufficient evidence in creation for us to know we are accountable to Him (vs. 19-20). Instead of glorifying God as Creator, mankind has chosen to worship all aspects of His creation (vs. 21-23). Because of this rejection of God, He has chosen to give mankind over to their own desires as a further downward spiral. He removes restraints over our sinful desires, so that our true sin nature is revealed. It is a worldview text that explains both the Roman culture and our own.

Paul is saying in these verses that the intensified practice of sexual immortality and idolatry IS the judgement of God (in contrast to God judging us for these deeds). Only the gospel is the power of God to rescue us from His judgment, which is why Paul is not ashamed of it (vs. 16). This is Paul's argument.