Catholics believe that Jesus gave special authority to his Apostles to rule and guide the Church. These Apostles then ordained and passed on authority to others, called bishops (literally ‘overseers’ in Scripture, it’s the same word). These bishops have ordained and passed on authority to other bishops, and so on, all the way up to the Church’s current bishops.
This is called Apostolic Succession, and it is absolutely essential to the constitution and life of the Church. What makes a bishop a bishop is if they have been consecrated by a bishop who was consecrated a bishop, all the way back to the Apostles who received authority from Jesus. A priest is only a real priest if he has been ordained a priest by a real bishop. If there is a break at some point – if a bishop wasn’t consecrated properly, or was consecrated by a bishop who wasn’t properly consecrated himself – then the line of authority stops at that point. It only works if there is no break in the line going all the way back to Jesus, the ultimate source of all authority in the Church.
Of course, this only works if Jesus did in fact give authority to his Apostles in the first place, and if his Apostles did in fact consecrate bishops with the instructions to pass their authority on to others in perpetuity. If the idea of apostolic succession was made up at some later point, and so did not originate with Jesus, then apostolic authority and succession isn’t real.
Most Protestants deny apostolic succession exists or is necessary for the Church. But so what? Why does apostolic succession matter? Here three ways:
1) Doctrine: The entire basis of the bishops’ teaching authority in the Church is the apostolic authority they’ve received from apostolic succession. Christians are obliged to follow the teachings of the Magisterium (the college of bishops headed by the bishop of Rome, the Pope) not because the bishops are smart, educated, or holy (some bishops are, but certainly not all), but because they have authority that ultimately comes from Jesus to teach in the Church and definitively interpret the deposit of faith.
If the bishops are not really bishops and do not have authority from Jesus, then they are simply one voice among many – there’s no reason anyone has to listen to them any more than anyone has to listen to the opinions of other theologians or preachers. But if they do have authority from Jesus, as Catholics claim, then they really do have a special charism of the Holy Spirit to protect them from error when definitively teaching the faith, and all Christians would have a moral obligation to follow their teachings – or else be heretics.
2) Worship: Certain Sacraments can only be validly performed by a bishop or a priest (a priest has been ordained by a bishop and has some of the powers of a bishop). The Sacraments that can only be performed by bishops and priests are Confirmation, the Eucharist, Reconciliation, and Anointing of the Sick. Holy Orders can only be performed by a bishop. (Baptism can be performed by deacons as well under ordinary circumstances and laypeople in emergencies, and in Holy Matrimony the spouses marry each other.) So, for example, if a layperson tried to consecrate the Eucharist, nothing would happen: transubstantiation would not occur, the bread and the wine would remain simply bread and wine.
So the validity of those five Sacraments listed above rests entirely on the veracity of apostolic succession. If the priest at your parish is not really a priest (either because he wasn’t validly ordained or because apostolic succession is false), the Eucharist is just bread and wine and your sins are not being forgiven in Reconciliation. But if apostolic succession is true, and our bishops and priests are real bishops and priests with the indelible mark of Holy Orders on their souls, then those Sacraments are truly effective – and in fact, necessary to the Christian life.
3) Unity: Bishops not only have authority to teach and perform the Sacraments, they also have authority to govern the Church and they serve as visible markers of the Church for unity. In other words, you can know that you are fully a member of the Church is you are in communion with a bishop who is in communion with the bishop of Rome. The sin of schism is when a baptized person intentionally breaks from the bishop of Rome and the bishops in communion with him.
But again, this is only true if the bishops really are bishops and have apostolic authority from apostolic succession. If they are not, if apostolic succession is false, then there’s nothing special about them and you don’t have to follow them.
So you can see, if apostolic succession is false, then the Catholic Church is largely a sham and Protestants are right. Everything salient about the Catholic Church stands or falls on the reality of apostolic succession.