This has to do with how God is able to save sinners without compromising what He is in Himself. “The righteousness of God” is about God’s acting in love to save sinners, but at the same time, not compromising what He is as a holy and just God (Rom. 3:21).
Man’s sin has seemingly put God in a dilemma. Since “God is love” (1 John 4:9), His very nature calls for the blessing of man, for He loves all men (John 3:16). But at the same time, “God is light” (1 John 1:5), and thus His holy nature justly demands that man should be judged for his sins (Heb. 2:2). If God acted according to His heart of love and brought men into blessing without dealing with their sins, He would cease to be holy and just. On the other hand, if God acted according to His holy nature and judged men according to the claims of divine justice, all men would be justly sent to hell, and none would be saved—and the love of God would remain unknown. How then can God save men and still remain just? The gospel announces this. It declares God’s righteousness and reveals the good news that He has found a way to meet His holy claims against sin and thus be able to reach out in love to save sinners who believe. Thus, God is presented in the gospel as being “just and the Justifier of him which believeth in Jesus” (chap. 3:26).
Many Christians have the idea that God's righteousness is something that is imparted to, or bestowed upon, or given to, the believer. However, Scripture does not present it in that way. Simply put, the righteousness of God refers to an act of God, not a commodity that He passes to men when they believe. If God gave His righteousness to us when we were saved, then He would no longer have it! W. Scott said, "It is God's righteousness, not man's. God cannot impute that which is essential to Himself in His dealings with men" (Unscriptural Phraseology, p. 10). It is true that God has given righteousness (Rom. 5:17), but this is in the sense of having secured or provided it for mankind in Christ, the risen and glorified Man. Thus, Christ has been "made unto us wisdom and righteousness" (1 Cor. 1:30), and we have been "made the righteousness of God in Him" in testimony to the world (2 Cor. 5:21). But the righteousness of God is not something that has been sent down from heaven and placed upon the believer, as some think.
W. Scott also said, “It is not putting a quantity of righteousness in, or on, a man" (Doctrinal Summaries, p. 15).
J. N. Darby remarked, “A man's being reckoned righteous is his standing in the sight of God, not a quantum of righteousness transferred to him” (Collected Writings, vol. 23, p. 254).
F. B. Hole said something similar: “We must not read those words [“the righteousness of God”] with a commercial idea in our minds, as though they meant that we come to God bringing so much faith for which we receive in exchange so much righteousness, just as a shopkeeper across the counter exchanges goods for cash” (Outlines of Truth, p. 5).
Some modern translations unfortunately say, "a righteousness from God" (Rom. 1:17; 3:21; 3:22; 10:3), or "the righteousness that comes from God" (Phil. 3:9). These are mistranslations and they have caused people to think that it is something that God imparts, or gives, to believers when they believe the gospel.
The righteousness of God points to what God has done in taking up the question of sin and settling it for His own glory and for the blessing of man. He sent His Son to be the Sin-bearer, and in His death, God judged sin according to His holiness. The Lord Jesus took the believer's place before God and bore his sins (the judgment of them) in His "own body on the tree" (1 Peter 2:24). His "finished" work on the cross (John 19:30) has rendered a full satisfaction to the claims of divine justice and has paid the price for the believer's sins. Thus, God has not compromised what He is as a holy and just God in reaching out in blessing to man. God's love has come out to men with the good news that He can, on a righteous basis, redeem, forgive, justify, and reconcile the sinner who believes.