For the love of Christ constrains us (‘grips us tightly'), because we thus judge, that one died for all, therefore all died, and he died for all, that they who live should no longer live to themselves, but to him who for their sakes died and rose again.
For what they do, they do because they are constrained by the love of Christ, the love that Christ has for them (it could mean the love that we have for Christ, but the immediate reference to the cross points to His love for us). They are gripped by His love. His love for them, revealed through the cross, moves them to reveal a similar love for others. Was He willing to die for them? So were they willing to die for others. Did He show His love for them? So will they show their love for others.
Indeed the death of Christ was such that they ‘all' partake in it. He died ‘for all' (that is for all who believe, whether Jew or Gentile), and 1 Corinthians 15:3 tells us it was ‘for our sins'. And in that fact that He died for all, all died. His death for sins was counted on their behalf. The fact that the latter ‘all' must refer only to Christians suggests that the first does also.
So the dying figure on the cross suffered for the sins of all who would be His. And as He died, we died in Him. His death comprised in itself a multiplicity of deaths, the deaths of all who would be ‘in Him'. The sentence of death on sin was being paid in Him for that innumerable multitude. That this has substitutionary force cannot be reasonably denied, although we can also include representation. He died in their place and as their representative, and thus they consider themselves as having died with Him (Galatians 2:20). His death is put to their account so that the law cannot condemn them. It has been satisfied by their having died in Him (Galatians 3:10-13) and it can no longer point the accusing finger (Romans 7:6). For if it did we would boldly reply, ‘I have already died in Christ. The price I owe has been paid.'
And the final purpose in His dying for all was so that those who did die with Him may no longer live to themselves, but to Him Who for their sakes died for them and rose again. They are to consider themselves, as they once were, as ‘the old man', as having died so that their lives no longer belong to them. They must reckon themselves as dead to sin and alive to God (Romans 6:11). And they must reckon themselves, as they are as the new man, as having risen with Christ, and therefore as being under obligation to God to live as He lives. For they have been raised in Him into heavenly places (Ephesians 2:6) and must live heavenly lives as citizens of heaven (Philippians 3:20).
The further significance of the cross is that those who come to receive the benefit of it in forgiveness of sin and in salvation (‘for our sins'), then recognise that as He died on the cross so did they, and they therefore recognise that being dead to sin they must live as dead to sin. They must die to all that put Christ on the cross. They must crucify the flesh with its affections and desires (Galatians 5:24). And they must see themselves as having risen in Him to a new life, so as to please the One Who Himself also died and rose again for their sakes. They must let Him live through them. In the words of Paul elsewhere, “I have been crucified with Christ. Nevertheless I live. Yet it is not I who live, but Christ Who lives in me. And the life that I now live in the flesh I live by faith in the Son of God Who loved me and gave Himself up for me' (Galatians 2:20). He recognised that Christ was now living in him, and desired to live through him. Thus his life from that time was a life offered to the One Who loved Him. This is why the Corinthians can recognise the genuineness of his message and of his concerns.