The doctrine that "Outside the Church there is no salvation" is one that is constantly misinterpreted by those who won't submit to the Magisterium of the Church. Faith does not depend upon our ability to reason to the truth but on our humility before the Truth presented to us by those to whom Christ entrusted that task. This is why the First Vatican Council taught that it is the task of the Magisterium ALONE to determine and expound the meaning of the Tradition - including "outside the Church no salvation."
Concerning this doctrine the Pope of Vatican I, Pius IX, spoke on two different occasions. In an allocution (address to an audience) on December 9th, 1854 he said:
[I could not find the text of the above document, but here's a document given March 17, 1856:We must hold as of the faith, that out of the Apostolic Roman Church there is no salvation; that she is the only ark of safety, and whosoever is not in her perishes in the deluge; we must also, on the other hand, recognize with certainty that those who are invincible in ignorance of the true religion are not guilty for this in the eyes of the Lord. And who would presume to mark out the limits of this ignorance according to the character and diversity of peoples, countries, minds and the rest?
Do not cease to diligently defend your people against these pernicious errors. Saturate them with the doctrine of Catholic truth more accurately each day. Teach them that just as there is only one God, one Christ, one Holy Spirit, so there is also only one truth which is divinely revealed. There is only one divine faith which is the beginning of salvation for mankind and the basis of all justification, the faith by which the just person lives and without which it is impossible to please God and to come to the community of His children. There is only one true, holy, Catholic church, which is the Apostolic Roman Church. There is only one See founded in Peter by the word of the Lord, outside of which we cannot find either true faith or eternal salvation. He who does not have the Church for a mother cannot have God for a father, and whoever abandons the See of Peter on which the Church is established trusts falsely that he is in the Church. Thus, there can be no greater crime, no more hideous stain than to stand up against Christ, than to divide the Church engendered and purchased by His blood, than to forget evangelical love and to combat with the furor of hostile discord the harmony of the people of God. Singulari quidem, March 17, 1856 (emphasis mine).
Again, in his encyclical Quanto conficiamur moerore of 10 August, 1863 addressed to the Italian bishops, he said:
These statements are consistent with the understanding of the Church contained in the documents of Vatican II, and the Catechism of the Catholic Church, as well as explaining why the rigorist position of Fr. Feeney (that all must be actual members of the Catholic Church to be saved) has been condemned by the Magisterium. It is ironic that precisely those who know their obligation to remain united to the Magisterium, and thus on whom this doctrine is morally binding, keep themselves from union with the Roman See on this point.It is known to us and to you that those who are in invincible ignorance of our most holy religion, but who observe carefully the natural law, and the precepts graven by God upon the hearts of all men, and who being disposed to obey God lead an honest and upright life, may, aided by the light of divine grace, attain to eternal life; for God who sees clearly, searches and knows the heart, the disposition, the thoughts and intentions of each, in His supreme mercy and goodness by no means permits that anyone suffer eternal punishment, who has not of his own free will fallen into sin. [Emphasis mine].
Answered by Colin B. Donovan, STL
Emphasis and added links by Scott Windsor