The most well-known reason is that since there is an additional dimension of Happiness on Holidays and we are less distracted from the travails of making a livelihood the Kohanim are able to bless the people.
It is interesting to note that over the generations certain Torah Sages, most notably the Vilna Gaon, believed the blessing should be said daily worldwide. They tried to institute this custom but were always unsuccessful. However, the Vilna Gaon’s students who moved to Israel succeeded in instituting the custom there. In addition, some Sefardic communities have Birkat Kohanim every day and there are other Sefardic communities that have it every Shabbat but not during the week.
An interesting note: HaRav Aharon Leib Steinman, a great Rabbi living in Israel today, prays at a very early Minyan. When he visits America, after his early weekday prayer, asks to be taken to a Minyan that has Birkat Kohanim so that he can be blessed!
