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.
