Generally speaking, the word flesh often takes on a negative connotation in Scripture, though not always. In this context, the flesh would be referring to those things which oppose the Holy Spirit of God. We know what is right and good and yet may fail to do it when the flesh interferes. If A is attempting to help B see the workings of the flesh in B's life, and B resists beyond the point where A sees they are making no headway, then turning B over for the destruction of their flesh simply means A leaves it in God's hands to deal with, in whatever way He sees fit.