According to the Bible, the Church Jesus established is the pillar and foundation of the truth; the gates of hell will not prevail against it; and the Holy Spirit will lead it to all truth. Toward this end, the Holy Spirit preserves the leaders of His Church from the possibility of teaching error, which is the definition of infallibility. It makes perfect sense that for us to know the truth of the Word of God, someone needs to have this charism. It accompanies both the office of the Pope and the body of bishops when they speak in their official capacity to the universal church regarding faith and morals.
Infallibility belongs in a special way to the Pope as head of the bishops (Matt. 16:17–19; John 21:15–17). Vatican II states it is a charism the Pope "enjoys in virtue of his office, when, as the supreme shepherd and teacher of all the faithful, who confirms his brethren in their faith (Luke 22:32), he proclaims by a definitive act some doctrine of faith or morals." Although the individual bishops do not enjoy the prerogative of infallibility, they can nevertheless proclaim Christ’s doctrine infallibly. Vatican II informs us that the bishops, "when gathered together in an ecumenical council, are teachers and judges of faith and morals for the universal Church. Their definitions must then be adhered to with the submission of faith" (Lumen Gentium 25).
Like many of the teachings of the Catholic Church, infallibility is one that is often misunderstood and then criticized or rejected based on that misunderstanding. It does not mean that the leaders are without sin, can create or change doctrine, are inspired or infused with knowledge of all truth, or can predict future events. And it's only in regard to faith and morals. Not physics, astronomy, or any other sciences, personal opinions, or comments given in a press conference.
It's interesting that the Catholic Church is the only one that claims to have infallible authority. But it's also proper, since Christ established one Church, put someone in charge, commissioned it to make disciples of all nations, and promised to be with it always along with the Advocate He would send to lead it to all truth (Matt. 28:20, John 14:16-17, 16:13).
Infallibility is sensible and necessary because the Church was established by God and guarantees the truth of His Word. We need to abide by this promised charism as proclaimed and found today only in the Catholic Church.