Q: Why do Catholics hold hands while reciting the "Our Father"?
A: Ooo, I actually know the answer to this question! Hand holding during mass is a practice that American Catholics adopted in the 1960's. It was one of many things they did, in the hope that it would make Catholic masses more appealing to Protestants.
Specifically, hand holding comes from the Quakers. They hold hands while they pray, as a symbolic gesture. It indicates that they are all connected through prayer, both metaphorically and physically. The practice developed, as a result of Quakers not having priests. They wanted to have a way to strengthen their prayers, without the use of priests; they decided that hand holding and communal prayer would work.
I should note that only American Catholics hold hands during the "Our Father". Catholics in other countries don't follow this practice. If you try to hold hands with them, they will look at you strangely because they have no idea what you're doing. Also, if you read the Roman Missal (which details the rubrics for celebrating mass), it says nothing absolutely nothing about hand-holding there, either for or against it.