Justification: Christ Alone
The basis for our justification is the grace of God in Jesus Christ, not how much faith we have. The ground of our justification is in Christ, and thus it does not depend on us, and not on our works. The reason why we are justified by grace alone through faith alone is because justification is in Christ alone. The ground of salvation is outside of us (extra nos) and therefore when the Scriptures say that salvation is a free gift, it really is a free gift, since nothing we do will ever affect the salvation we receive by faith alone.
The focus on Christ alone is made in contrast to two errors. The first error is to focus on the will of men. This was the problem with the early 17th century Arminians, and the problem with many of the 15h and 16th century Renaissance humanists like Desiderius Erasmus (and with Rome too). In such systems, it is Christ and us who co-operate to save ourselves. God plays his part, even most of the work, but it is now up to the individual person to take the last step towards salvation. The second error is to supplement Jesus with the cult of the saints, especially Mary, which was prevalent and still is present in Roman Catholicism. Prayers to various saints and especially Mary are perceived to help supplement one's devotion towards God. In the case of Mary, outrageous claims have been made in Roman Catholicism about the simple girl from Nazareth, to elevate her to a person with almost god-like status. Prayers to the saints, and especially to Mary, are encouraged in Roman Catholicism, even with helping out in one's standing before God. But if justification is in Christ alone, then the saints and Mary (assuming they can hear our prayers, and that without being appalled by being treated as an object of prayer) cannot help at all. Christ is the only mediator between God and men (1 Tim. 2:5) and there cannot be any other, neither "the saints" nor Mary.
Christ alone saves because He is the only righteous man. His righteousness can be split into His passive and His active righteousness, both of which are given or reckoned to us.
Christ's passive righteousness and satisfaction
Christ's passive righteousness refers to the righteousness of God that comes from His suffering. It is not "passive" in contrast to active, but rather "passive" as in full of passions. The sufferings of Christ begin from the time the Son become incarnate, took on a human body, and had to limit Himself to a human with all our limitations, and culminates at the Cross, where Christ suffered, bled and died on the Cross.
This passive righteousness of Christ, as focused on the Cross, is tied up with the notion of satisfaction, which is the doctrine that Jesus died to satisfy or quench or propitiate the wrath of God. God is angry against sin, and His wrath burns against sinners. It was because of the universal knowledge of God's wrath that ancient cultures around the world, distorting the original knowledge of God, had animal and even human sacrifices to propitiate the wrath of their gods. In Israel, God re-instituted the pure sacrificial system to teach Israel and us that, apart from the shedding of blood, there is no forgiveness of sins (Heb. 9:22).
Far from being an invention of backward savages, the bloody sacrificial system points to the truth that the shedding of blood as payments for sins is necessary to return to a right relationship with God. Jesus by dying on the Cross paid the price for our salvation so that we do not need to shed any more blood to be justified.
Christ's active righteousness and imputation
Christ's active righteousness consists in his obedience to the demands of the law of God. It is active because it consists in the actions of obedience. As Jesus perfectly obeys the law and obeys the will of the Father, He merits life, the only human to ever have done so (Rom. 5:16b, 17b). Jesus' perfect obedience shows that He is supremely worthy of eternal life, and this life He gives to us His people who believe in Him.
Jesus gives us His righteousness through imputing or reckoning it to us. Imputation is the doctrine that states that we are given Jesus' righteousness so that we are seen as if we are actually righteous even though we are not in nature and action. As Scriptures states, "For our sake he made him to be sin who knew no sin, so that in him we might become the righteousness of God." (2 Cor. 5:21). Our sins are imputed to Christ, while Christ's righteousness is imputed to us. Jesus was treated as if He were sinful even though He was not, and we who believe in Christ are treated as if we are righteous even though we are not.
This notion of double imputation, or the Great Exchange, is the crux of our salvation. The reason why we can be justified by grace alone through faith alone apart from works is because of what Christ has done on our behalf (pro nobis). We are justified because Jesus has given us His righteousness, not just paid our punishments but that we have actual righteousness credited to our account. When God the Father sees us, He sees the righteousness of Christ, not our ugly sins, because of what Jesus did for us.
The eternal and glorious God, without any compulsion, decided to save His people, and He does this through the life and work of our Savior Jesus Christ. We are regenerated by the Holy Spirit, then justified by grace alone through faith alone in Christ alone. Jesus took our sins and appeased God's wrath, and imputed to us His righteousness, and thus make us right before God and pleasing to Him.