Then why Jesus died for my sins? Why i must repent and confess? Am i responsible here for sins of my father/s?
There are different theological streams on this issue. Broadly they are referred to as "Atonement Theories", attempts at bringing together various theological components surrounding Jesus' death and putting them into a cohesive and meaningful whole.
The New Testament itself doesn't do this so much (though many would disagree with me on that point); rather the New Testament uses a lot of imagery to try and crystallize the monumental importance Jesus' death played in history and redemption. It draws from the imagery of the Paschal Lamb, it draws from the imagery of Jewish sacrifices, it draws imagery from Adam, and so on and so forth.
One of the earliest theories put forth can be found in the writings of St. Irenaeus who draws heavily upon the writings of St. Paul describing Jesus as the "Second Adam", this is the essential imagery that Irenaeus uses. For Irenaeaus Jesus is the Second Adam, the New Man, who by His human life recapitulates (this is why it's called Recapitulation Theory) Adam's life, specifically by undoing Adam's disobedience with His own obedience. Thereby Jesus, by participating in total the full meaning of human existence, and sharing even in death, He can then by His resurrection restore and bring to new life the entire human race. If by Adam's disobedience sin and death entered the world, then by Christ's obedience He has put to death sin and rendered death powerless, thereby coming resurrection from the dead. It is for this reason that, being in Christ, we share in His resurrection; even as He was raised from the dead as first-fruits, so shall we at the consummation of history be raised up from the dead to everlasting life.
Later Fathers used the imagery of ransom. The idea is that by our sin we have become captive prisoners of the devil, God out of His great love for us offers His only-begotten Son as a ransom. The devil receives this payment, Christ dies but then God in a twist, raises Christ from the dead and thus rendering the devil powerless, defeated and empty-handed. We call this Ransom Theory.
No later in the 11th century, medieval theologian and philosopher St. Anselm writes a work titled
Cur Deus Homo?, or "Why God Became Man". In this work, Anselm draws upon the imagery of God as an offended lord, His honor offended by the sin of man and thus deserving restitution for the offense received. However man is incapable of paying this honor-debt to God, thus what is needed is one who can stand as God's equal and also act as representative of man: Jesus, being God-Man, alone can pay our debt on our behalf, therefore satisfying God's honor. We call this Satisfaction Theory.
St. Thomas Aquinas, a little later, adapts this but rather than God's honor being offended, it is God's justice.
During the Reformation, some Reformers such as John Calvin adapted Satisfaction Theory further, producing Penal Substitution Theory. Here we have trampled God's Perfect and Divine Law, and the penalty is death. Christ, therefore, pays the price on our behalf, vicariously (hence "Vicarious Atonement"), suffering the death we deserve and thus covering us with His righteousness. It's this view, through several permutations, that is the typical Atonement Theory and view one finds prominent among modern Evangelicals, as well as other Protestants.
About a century ago a Lutheran theologian, Gustav Aulen wrote a book titled
Christus Victor, wherein he challenged the tradition of Satisfaction Theory (from Anselm and beyond), and looks toward the ancient views of the Christian Church (Recapitulation and Ransom). A reiteration of these ancient views has since taken name from the title of Aulen's book, and has been called Christus Victor Theory. It's since had a growing popularity among Western Christians, many such as myself who feel that the Medieval and Early Modern views in the West fall short at really capturing the mystery and profundity of the Atonement; and that the views of antiquity address the issue in a much more profound way.
In other words, I more-or-less subscribe to Christus Victor Theory. Jesus doesn't die to pay a debt to God, with God's wrath pouring out on Him because of Adam. Rather, God in Christ comes down to restore and to reconcile us, and to put an end to all that hinders us from full communion with God and one another: namely sin, death and hell.
As new babies are born, does that mean Jesus died for their sins? Sins that they did not commit?
We confess that Christ died for the sins of the whole world.
Furthermore, in Western Christianity (it should be pointed out that the Eastern Church doesn't adhere to the Western concept of Original Sin), Original Sin isn't about making babies guilty for things they haven't done; rather it's about addressing the fundamental problem of sin. The infant isn't a sinner because they're evil and guilty; rather it's that each and every one of us are heirs of a human nature left depraved by sin.
The Eastern view is different. Rather than "Original Sin", the Orthodox typically speak of "Ancestral Sin", which more-or-less says that through Adam's disobedience the world itself has been held captive to sin and death. Thus it's less about inheriting a sinful nature from Adam in the Western sense, but more about that we are born into an environment that is systematically diseased and we thus it's inevitable that we are sick.
This distinction between East and West goes further into the very idea of Sin. In the West Sin is primarily viewed as a judicial problem and spoken of in judicial language: we have violated God's just law, punishment and restitution are needed. In the East Sin is primarily viewed as a medical problem and spoken of in medical language: we are sick, medicine and healing is what is needed. Medical language actually permeates Christian history. St. Augustine referred to the Church as a hospital for sinners. St. Ignatius of Antioch referred to the Eucharist as medicine of immortality and the antidote against death. The role of the Church therefore is to administer medicine and healing to a sick world and care for them as representatives and ambassadors from Christ, the Great Physician.
-CryptoLutheran