1 What I am saying is that as long as an heir is underage, he is no different from a slave, although he owns the whole estate. 2 The heir is subject to guardians and trustees until the time set by his father. 3 So also, when we were underage, we were in slavery under the elemental spiritual forces of the world. 4 But when the set time had fully come, God sent his Son, born of a woman, born under the law, 5 to redeem those under the law, that we might receive adoption to sonship. 6 Because you are his sons, God sent the Spirit of his Son into our hearts, the Spirit who calls out, “Abba, Father.” 7 So you are no longer a slave, but God’s child; and since you are his child, God has made you also an heir. (NIV)In this section Paul is reiterating the argument of the last section. This repetition signals that this is a critical portion of his argument (Martyn goes as far as to call it the most important section of the letter). The pressing question throughout Galatians is, 'who are the people of God?' The answer over and over again is the people who possess the Holy Spirit. This passage gives us a clearer picture of why Paul can make that claim.
The opening two verses of the passage provide an example that forms the basis of the analogy of verses 3-7. Martyn (385-6) has a very helpful chart that I will replicate below that is helpful for seeing how Paul's argument works.
|the heir in a household||we human beings|
|(v 1) as a child||(v 3) as children|
|(v 1) the heir is a virtual slave||(v 3) we were enslaved|
|(v 2) until the time set by the father||(v 4) but when the fullness of time came|
|(v 4) God sent his Son|
|(vv 1-2) for his transition out of virtual slavery into active lordship.||(v 5) to bring about our transition, by delivering us from slavery.|
|(v 5) we receive adoption as sons|
|(v 1) heir||(v 6) and you, as you are sons, God also sent into our hearts the Spirit of his Son, crying out, "Abba, Father"|
|(v 7) Thus, you are a son; and if a son, then also an heir by God's act of redemptive adoption.|
Not only does this chart provide us with a clear presentation, I also appreciate how it demonstrates the eschatological underpinnings of Paul's argument. When Jesus came we had a turning of the ages. Gone is the prior age of enslavement, ushered in is the time of blessing. Thus Paul is arguing that to go back to following the law is to go back to a less desirable situation because living in the sphere of the law is enslaving (the opposite of liberating). In fact in the case of the Jews, liberation from the sphere of the law was what they were waiting for. It is this liberation, that then is extended to all of us.  When we put the pieces of this passage together we see that we have a two stage act of redemption. Phase one was God's sending of the Son, phase two is the sending of the Spirit to God's redeemed people.
What ties it all together is the recognition that, the Spirit is the Spirit of the Son, Jesus' Spirit. The Spirit that enables us to cry out to God in prayer in imitation of Jesus personal prayer. The Spirit in us shows that we are 'in Christ' and hence members of the people of God.
 The most difficult exegetical decision that one faces in this passage is the question of the identity of the pronouns. Specifically, who are the we in verses 3 and 5? While it is tempting to see the 'we' as embracing both Jewish and Gentile Christians, the strong parallelism with Galatians 3:13-14 (see my post on those verses) inclines me to restrict the 'we' to Jewish Christians and see Paul's statements as salvation historical. At the same time, though, Paul's pronoun usage here is a bit messy, and it's clear that there's a strong emphasis on the extension of redemption to the Gentile audience of Galatia. So I won't quibble with those who want to take the 'we' wider (as e.g., Dunn does) as long as we notice that, 'Paul does not achieve universality of effect by abandoning historical particularity' (Dunn 216). One might object that the 'we' in vs. 5 is in contrast with 'those,' who are clearly Jewish Christians. The issue here is that if you take that position seriously then you need to hold that 'we' are Gentile Christians only. I think it's best to see the messiness with the pronouns there stemming from that fact that verse four and the first half of verse 5 is part of a creed-esque statement from the early church that Paul is quoting. The second half of verse 5 begins Paul's application of that piece of tradition.