It is not a prescribed part of the liturgy, and appears to be an accretion from Protestant tradition. One authoritative source says
To find the rubrics (regulations which govern the Mass) concerning these gestures, one may turn to the General Instruction of the Roman Missal (1970), On Holy Communion and the Worship of the Eucharistic Mystery Outside of Mass (1973), Instruction on the Worship of the Eucharistic Mystery (1980), and Instruction on Certain Norms Concerning the Worship of the Eucharistic Mystery (1980).
However, in all of the liturgical documents for the universal Church or of those particular ones issued by the United States Bishops Conference, no where is the holding of hands during the Lord's Prayer mandated. Frankly, this gesture arose among the various liturgical innovations in the aftermath of Vatican Council II.
He goes on to surmise (probably accurately)
Perhaps the holding of hands was introduced with good intentions to highlight the unity of the congregation as they pray, "Our Father," not "My Father."
As you say, there are those whose attitude towards the practice ranges from uneasiness to revulsion, and for those people the practice must be counter-productive!
That same commentator goes on to explain why it's not even necessary.
The unity that is sought really comes later and after a spiritual progression: First, we fall on our knees as the priest offers the sacrifice of the Mass: we recall not only our Lord's passion, death, and resurrection but also our need as individuals to offer ourselves to Him. Second, we pray in the words our Savior taught us, the Lord's Prayer, in which we ask, "Forgive us our trespasses as we forgive those who trespass against us," even the person next to us in the pew. Third, we offer the Sign of Peace, a gesture found in the earliest Masses to show a genuine unity based on peace and forgiveness. Finally, we receive Holy Communion, which truly brings us into communion with our Lord and with each other. Looking at the logic of this spiritual progression to real unity, the holding of hands at the Our Father is extraneous.
At least one bishop is reported to have explicitly discouraged the practice:
WCPO reports that Bishop Foys [of Covington KY] wrote:
Special note should also be made concerning the gesture for the Our Father. Only the priest is given the instruction to "extend" his hands. Neither the deacon nor the lay faithful are instructed to do this. No gesture is prescribed for the lay faithful in the Roman Missal ; nor the General Instruction of the Roman Missal , therefore the extending or holding of hands by the faithful should not be performed.