The ESV MacArthur Study Bible says the following:
2 Cor. 5:14 the love of Christ. Christ’s love for Paul and all believers at the cross (cf. Rom. 5:6–8). Christ’s loving, substitutionary death motivated Paul’s service for him (cf. Gal. 2:20; Eph. 3:19). controls. This refers to pressure that causes action. Paul emphasized the strength of his desire to offer his life to the Lord. one has died for all. This expresses the truth of Christ’s substitutionary death. The preposition “for” indicates he died “in behalf of,” or “in the place of ” all (cf. Isa. 53:4–12; Gal. 3:13; Heb. 9:11–14). This truth is at the heart of the doctrine of salvation. God’s wrath against sin required death; Jesus took that wrath and died in the sinner’s place. Thus he took away God’s wrath and satisfied God’s justice as a perfect sacrifice (see notes on 2 Cor. 5:21; Rom. 5:6–11, 18–19; 1 Tim. 2:5–6; cf. Eph. 5:2; 1 Thess. 5:10; Titus 2:14; 1 Pet. 2:24). therefore all have died. Everyone who died in Christ receives the benefits of his substitutionary death (see notes on Rom. 3:24–26; 6:8). With this short phrase, Paul defined the extent of the atonement and limited its application. This statement logically completes the meaning of the preceding phrase, in effect saying, “Christ died for all who died in him,” or “One died for all, therefore all died” (see notes on 2 Cor. 5:19–21; cf. John 10:11–16; Acts 20:28). Paul was overwhelmed with gratitude that Christ loved him and was so gracious as to make him a part of the “all” who died in him.[2]
The HCSB Study Bible says about 2 Corinthians 5:13-15:
...5:13-14 Paul's opponents probably had suggested that he was religiously unbalanced (see Ac 26:24). He was "insane" in that Christ's love compelled him into vigorous apostolic ministry. On the other hand, his ministry among the Corinthians had never been that of a madman (1Co 2:1-5). Indeed, he had kept his "third heaven" vision private for 14 years until he mentioned it later in this letter (12:1-10). The heart of Paul's message was that the Jewish Messiah had died on behalf of all kinds of sinners (1Co 15:3). Jews as well as Gentiles were included in Jesus' substitutionary death (Rev 7:9). In union with Christ, sinners who believe the gospel have died to sin and have been raised to walk in a new way of life.
5:15 The phrase those who live refers to believers who are now spiritually alive (Eph 2:4-6). Christ's death a