To be fair, the infallibility of the Church in general has never been given an extraordinary definition–it has always been the common teaching of the Church. The Pope’s infallibility is simply a function of that and his relationship to the rest of the Church. The Church always received the Pope’s irreformable judgments as irreformable (and therefore infallible).
However, in the mid-19th century, the Rationalists opposed the concept of faith, subjecting everything to reason alone. The Church, on the other hand, teaches revealed truth must be believed by faith. This only makes sense if what we are believing with faith cannot be subject to error (something subject to error, should not be given absolute faith).
Therefore the Church needed to reiterate its infallibility, and which of its judgments were infallible, in relation to its defense of the virtue of faith.
The First Vatican Council had planned to have more documents on the rest of the episcopate and Church, which would have talked more about their infallibility (this was later done by Vatican II). It just happened to start with the documents on faith and on the Pope, and then didn’t get to finish because Rome was invaded and taken over.
The Pope’s infallibility was implied in the fact that he is, in the words of the Council of Florence, “the father and teacher of all Christians, and to him was committed in blessed Peter the full power of tending, ruling and governing the whole church.”
It is also tied to the fact that the Roman Church–the Apostolic See–cannot defect from the faith and Church, as then the Church would lose the constituent element of the primacy.
As Pope St. Agatho I put it in the 7th century:
…from the beginning [the Roman See] has received the Christian faith from her founders, the princes of the Apostles of Christ, and remains undefiled unto the end, according to the divine promise of the Lord and Saviour himself, which he uttered in the holy Gospels to the prince of his disciples: saying, Peter, Peter, behold, Satan has desired to have you, that he might sift you as wheat; but I have prayed for you, that (your) faith fail not. And when you are converted, strengthen your brethren.
http://www.newadvent.org/fathers/3813.htm