Note: the following is not a defense of the doctrine, but an explanation of what it means to Catholics. You'll probably still disagree with the truth of the doctrine, but now at least you'll know what to actually disagree with:
First off, "Ex cathedra" or "from the chair" is a figure of speech--it doesn't mean he has to literally sit on a special chair. It is a figure of speech that means he is exercisng his authoirty over the whole Church (which is what the phrase "chair of Peter" is sumbolic of).
We as Catholics believe the Church will never lose the Truth. So any teaching that must be held by the entire Church must be faithful to the original deposit of faith delivered by the Apostles. Throughout history, doctrinal disputes have arisen. To settle those disputes, the Church will issue a definitve proclamation to be held by the entire Church as to what the Truth is on the matter. If the definitions were false, the truth will have been lost.
Since the Successor of St. Peter has the special ministry from God to make such definitive judgments (to confirm the brethren in the faith), his proclamations that are to be held by the entire Church are infallible. Even the definitions of ecumenical councils cannot be binding on the entire Church without his approval.
Infallibility is not inspiration--the Pope doesn't get new revelation from God like the LDS folks. he also doesn't automatically get the correct answer or the best way to explain it .in his head What it does mean is that the Holy Spirit, the Spirit of Truth who preserves the Truth in the Church, does not allow Him to definitvely bind the entire Church to a false doctrine.
Rather than exalting the Pope above God, the doctrine of Papal infallibility asserts God's sovereignty over the pope.