Why is Apostolic succession required in some denominations?

The power to transform bread and wine into the divine body and blood of Christ was transferred by Jesus Christ to the Apostles. This allows us today to truly receive Jesus Christ as the Most Holy And Perfect Sacrifice. This also includes other Sacramental powers i.e. Ordination, Baptism, Matrimony, Confirmation, Reconciliation, Anointing of the sick. It is through Apostolic succession that these Divine powers are transferred to the next generation and so on and so on, extending back to Christ Himself! The Church is the Divine instrument established on earth founded by Jesus Christ through the Apostles and their successors. The bible is one of many manifestation of the Church that would not have been established and preserved but by and for the Church under the divine guidance of God the Holy Spirit. The Church is what Jesus established as the new Covenant to replace the old Covenent and represents God's Kingdom on earth until such time that Jesus returns and the Eternal Kingdom free of sin and death is establised by God for all eternity! This is why Apostolic succesion is important.

As a protestant...Why do churches which hold to Apostolic succession require it?

The Christian churches that cling to Apostolic Succession, such as the Roman Catholic Church and the Orthodox Catholic Church, teach that the legitimate authoritative succession of bishops is taught in both Sacred Scripture and Sacred Tradition.

History also clearly indicates that Apostolic Succession (the doctrine itself - not the legitimacy of the successor) was virtually uncontested until the Protestant Reformation of the 16th. century.

The Catholic Church teaches that Apostolic Succession is necessary because it was instituted by the original Apostles. Other denominations claims are similar.

From the Catholic Encyclopedia:

It remains to consider whether the so-called "monarchical" episcopate was instituted by the Apostles. Besides establishing a college of presbyter-bishops, did they further place one man in a position of supremacy, entrusting the government of the Church to him, and endowing him with Apostolic authority over the Christian community? Even if we take into account the Scriptural evidence alone, there are sufficient grounds for answering this question in the affirmative. From the time of the dispersion of the Apostles, St. James appears in an episcopal relation to the Church of Jerusalem (Acts 12:17; 15:13; Galatians 2:12). In the other Christian communities the institution of "monarchical" bishops was a somewhat later development. At first the Apostles themselves fulfilled, it would seem, all the duties of supreme oversight. They established the office when the growing needs of the Church demanded it. The Pastoral Epistles leave no room to doubt that Timothy and Titus were sent as bishops to Ephesus and to Crete respectively. To Timothy full Apostolic powers are conceded. Notwithstanding his youth he holds authority over both clergy and laity. To him is confided the duty of guarding the purity of the Church's faith, of ordaining priests, of exercising jurisdiction. Moreover, St. Paul's exhortation to him, "to keep the commandment without spot, blameless, unto the coming of our Lord Jesus Christ" shows that this was no transitory mission. A charge so worded includes in its sweep, not Timothy alone, but his successors in an office which is to last until the Second Advent. Local tradition unhesitatingly reckoned him among the occupants of the episcopal see. At the Council of Chalcedon, the Church of Ephesus counted a succession of twenty-seven bishops commencing with Timothy (Mansi, VII, 293; cf. Eusebius, Church History III.4-5).

The Orthodox Wiki explains the significance and importance like this:

The unbrokenness of apostolic succession is significant because of Jesus Christ's promise that the "gates of hell" (Matthew 16:18) would not prevail against the Church, and his promise that he himself would be with the apostles to "the end of the age" (Matthew 28:20). According to this interpretation, a complete disruption or end of such apostolic succession would mean that these promises were not kept as would an apostolic succession which, while formally intact, completely abandoned the teachings of the Apostles and their immediate successors; as, for example, if all the bishops of the world agreed to abrogate the Nicene Creed or repudiate the Holy Scripture.

The LDS teaching has a similar ring, if not a similar line of Presidents. They believe that the current lines recognized by Catholic and Orthodox Churches are invalid because all other denominations are apostate, and the "true Church" was restored by Joseph Smith. A new line of succession began with him, but the concept of succession is the same.

The belief is that the line in the "restored" Church" is ordained and protected by God. Therefore it is by His will that succession is necessary, and by His will that it is protected.

Succession in the Presidency of the Church has been established by the Lord. The Church is never without inspired leadership, and there is no reason for speculation or controversy over who will become the next President of the Church. President Harold B. Lee (1899–1973) explained: “[The Lord] knows whom he wants to preside over this church, and he will make no mistake. The Lord doesn’t do things by accident. He has never done anything accidentally” (in Conference Report, Oct. 1970, 153; or Improvement Era, Dec. 1970, 127). President Ezra Taft Benson (1899–1994) taught that “God knows all things, the end from the beginning, and no man becomes president of the church of Jesus Christ by accident, or remains there by chance, or is called home by happenstance” (“Jesus Christ—Gifts and Expectations,” New Era, May 1975, 16–17).

That may seem a simplistic answer, but the idea is that authority must be granted, and that for authority to be granted, it must come from one who has the authority to grant it. The need for a line of succession is similar to the need for a central Church authority, which was asked about at What is the doctrinal or Scriptural basis for a central Church authority? The accepted answer there provides Scriptural support for both ideas.

Borrowing from JustinY's answer:

A central church authority is a necessary byproduct of believing that certain ordinances are necessary for salvation. Ordinances often have rules for who can be administered to as well as how it must be done. Church authority is a very orderly and efficient way to regulate those requirements. The Old Testament is an excellent example of this. Notice that churches who put more emphasis on ordinances also have more structure (Catholic, Orthodox, and LDS churches, as well as the Jewish faith). And it's not just a good way to regulate the ordinances, but also a good way to regulate doctrine.