I am a priest, in case you were wondering :).
The doctrine of “no salvation outside the Church” is actually rather simple, but it has to be understood correctly. I usually explain it like this. There are two aspects:
(1) God wants every man, woman, and child to be a full member of the Catholic Church, because that is the instrument that He has set up to save them. Thus, everyone has, per se, an obligation to be baptized and to be members of the Church in good standing. A person cannot be saved if he willfully and with full knowledge rejects Baptism or full communion with the Church and persists in that refusal at the time of death. (It works like any other mortal sin.)
Having said that, there are a number of factors that may make such communion impossible: for example, some persons may not yet have heard the Gospel; others might have heard it but have prejudices that make it impossible for them to accept the Faith (this would apply to non-Christians as well as to non-Catholic Christians); or else, having decided to become members of the Church, perhaps through sudden illness or another circumstance, they die before receiving Baptism.
There is a saying in morality, Ad impossibilia nemo tenetur: “No one is obligated to do the impossible.” Thus, God will not condemn someone who does not become a Catholic through no fault of his own. Such persons will be judged based on how well they followed the Natural Law and the dictates of their own consciences (and, if they are Christian, how well they have lived the Gospel, in accord with their knowledge of it).
(2) Everyone who is saved, regardless of his religious affiliation, does so through the action of the Church. If a Protestant or Orthodox, or even a Buddhist or an atheist, is saved, it is thanks to the action of the Church.
With non-Catholic Christians it is easier to see, because they have access to at least some of the sacraments (namely Baptism and Matrimony), but this is true in a mysterious way even for those non-Christians that are saved.
This is true because the only Savior is Jesus Christ: only the Incarnate Son can possibly redeem mankind. Buddha, for example, is powerless to do so, even if we hope for his salvation. The Church, moreover, is so united to Christ as to be like his Body. Thus, no one can be saved by Christ and not simultaneously by the Church.
So that is the answer: “outside the Church there is no salvation,” because, on the one hand, everyone is required to be a member of the Church (but of course God does not demand the impossible); and everyone who is saved–regardless of his formal religious affiliation–is saved by the Church.
I hope that helps!
Father Louis Melahn, L.C.