The Catholic teaching is that no man or woman is saved apart from the sacrifice of Christ. I cannot emphasize this enough. To be fair to the RCC, this is absolutely their stance. However, the current RCC position is that due to "invincible ignorance", it is still possible that non-Christians (and non-Catholic Christians as well) have a chance at salvation. I do not know if this is exactly how some Protestants (or Orthodox for that matter) who agree with the Catholic Catechism, that it is possible that some non-Christians might be saved, would describe the situation or not. Some probably would and others might not.
How did the "invincible ignorance" doctrine develop? Well, it seems to have started with a handful of Early Church Fathers who taught that some of the Pagans that they prayed for were saved. The doctrine did not go very far at all, at least not in the West, until the New World was discovered as having Native Americans who had never heard of the Gospel message. Then one of the Popes mentioned it in a non-infallible teaching way as a method for the Native Americans to have a chance at salvation. It was mentioned in the 1800's in a general manner, not specific to any particular group, but again not with the authority of Papal Infallibility. It was again mentioned when Father Feeney was excommunicated in the 1940's, but yet again, in a non-infallible teaching manner. Finally with the Vatican II documents in the 1960's, the RCC officially developed the "invincible ignorance" doctrine in more detail and apparently with the safeguard of Papal Infallibility (as the RCC teaches anyway), except that very conservative Catholics maintain that since the Vatican II Council was a Pastoral Council and not a Dogmatic Council, the documents on Religious Liberty and Ecumenism are not protected by Papal Infallibility and are not in accord with the past teachings of the Church.
As to how some liberal/moderate Protestants (and some Orthodox) would explain their positions, is more difficult to say. I have seen some state that they believe that all souls at the moment of death will be given a last chance to accept Jesus. This would obviously be most important for those who have lived and died without hearing the Gospel message. Many others simply say that God will show mercy upon whom He chooses and they are just unwilling to put God in a box, so to speak, and try and speak for God on the possibility of salvation for non-Christians (obviously despite the fact that the Bible seems to speak clearly on the matter). I suppose that a very small handful of others might feel that the Old Testament Covenant with the Jews is still valid and that this Covenant gives Jews a chance at salvation. Others probably believe that God will judge non-Christians by the lives they have lived in accordance with the light that has been given to them.
To the Eastern Orthodox, they view salvation a little different than most Protestants and Catholics. They believe that upon death, we receive a foretaste or Heaven or Hell, but not real Heaven or Hell until Jesus returns. It appears that a minority view within Orthodoxy goes along with what a minority of Early Church Fathers taught, namely, that prayers for the deceased still have the chance to bring them to salvation, even if they died without knowing Jesus. I emphasize that this appears to be a minority view among Orthodoxy. However, the Eastern Orthodox do pray once a year at Pentecost for the souls in Hades (presumably those encountering a foretaste of Hell, as the Orthodox believe) and they believe that these prayers will at least give said souls some comfort. Personally, as a Protestant, I find such a prayer to be a wonderful thing. I do not know if it does any good, but I see no harm in it at all.
So, in summary, this line of thinking probably began as a way to explain how God could offer salvation to those who live and die without hearing the Gospel message.