I tend to think it implies the making of a conscience, reconciled, and finalized choice to trust in the finished work of Christ for one's righteousness before the Lord, and to abandon all other trusts in one's own works or merits. It is a way of expressing the making of a final decision. Like, whenever one decides to marry a person and actually becomes engaged, they "close" with that choice they selected and renounce all other options. We "close" on a house when we sign the final paper indicating the seriousness of our selection. And so, closing with Christ indicates that we have come to a place where we identify with him and him alone to stand for us before the Father as our only righteousness and assurance.
Blessings!