2 Corinthians 6:14 and its subsequent verses seem like something of an abrupt interlude in the middle of Paul's appeal to the Corinthians to open their hearts to him and his associates. What's the intended application here - how does separating themselves from unbelievers relate to the immediate context of Paul appealing to them over their closed hearts? In essence, how are verses 14-18 intended to strengthen his surrounding argument?
v14-18 is the section between the first and second bold highlights below:
2 Corinthians 6:11-7:2 (ESV)
We have spoken freely to you, Corinthians; our heart is wide open. You are not restricted by us, but you are restricted in your own affections. In return (I speak as to children) widen your hearts also. Do not be unequally yoked with unbelievers. For what partnership has righteousness with lawlessness? Or what fellowship has light with darkness? What accord has Christ with Belial? Or what portion does a believer share with an unbeliever? What agreement has the temple of God with idols? For we are the temple of the living God; as God said,
“I will make my dwelling among them and walk among them, and I will be their God, and they shall be my people. Therefore go out from their midst, and be separate from them, says the Lord, and touch no unclean thing; then I will welcome you, and I will be a father to you, and you shall be sons and daughters to me, says the Lord Almighty.”
7:1 Since we have these promises, beloved, let us cleanse ourselves from every defilement of body and spirit, bringing holiness to completion in the fear of God. Make room in your hearts for us. We have wronged no one, we have corrupted no one, we have taken advantage of no one.
Related question: 2 Corinthians 6:14 - What Does "Unequally Yoked" Mean?