The source for mentioning dreams in Birkas Kohanim comes from the Gemara in Berachos 55b
האי מאן דחזא חלמא ולא ידע מאי חזא, ליקום קמי כהני בעידנא דפרסי ידייהו ולימא הכי
The William Davidson (Koren Steinsaltz) translation:
One who saw a dream and does not know what he saw should stand before the priests when they lift their hands during the Priestly Blessing and say the following:
The Gemara then enumerates the prayer that one says during the Priestly Blessing. The Zohar (Part 3 Page 147b) also mentions this custom, and it is cited in the Tur and the Shulchan Aruch (Siman 130).
The Noam Elimelech by Rav Elimelech of Liznesk in Parshas Teruma (ד"ה וזה רמז למה, second column, 6th line from top) states:
על דרך דאיתא בספר למה שאנו אומרים בשעת ברכת כהנים רבונו של עולם חלום חלמתי, הטעם שהחלום הוא אחד מששים בנבואה, והכהנים הוא כולו נבואה שמטעם זה אסור להסתכל בידי הכהנים שהשכינה שורה בידיהם ולכן אנו אומרים רבונו של עולם כדי לבטל החלום בששים
My rough summary/translation:
"We say a prayer for dreams during the Priestly blessing because a dream is one sixtieth of prophecy, and the Priests achieve full prophecy during their blessing, and the dream becomes nullified during the Priestly blessing. For this same reason we are not allowed to look at the hands of the Priests because the Divine Presence rests among them."
The Noam Elimelech writes that Kohanim achieve full prophecy during the Birchas Kohanim, and since dreams are considered 1/60th of a prophecy, the dreams are nullified within the Priestly blessing according to the principle of בטל בשישים - batel bshishim, (from Wikipedia: "nullified in sixty; that is, permissible so long as forbidden ingredients constitute no more than 1/60 of the whole", see this article, especially the introduction and paragraph I for more details about this Halachic principle).
The idea that dreams are 1/60 of prophecy comes from the Gemara in Berachos 57b which states:
חלום אחד מששים לנבואה
The William Davidson (Koren Steinsaltz) translation:
A dream is one-sixtieth part of prophecy.
See this article for more information on this concept.
This works well with the 60 words drash that paquda cited in his answer.
EDIT: I seem to have found an earlier source for the Noam Elimelech's explanation. It is in the Sefer Toras Chaim (.ב"ק נה) by Rabbi Avraham Chaim Shorr (towards the bottom of the first column). I suggest reading it as it gives slightly more context than the short paragraph in Noam Elimelech. He mentions that a Birkas Kohanim is considered prophecy from God because the Kohanim are saying exactly the same words that God commanded to be said, as well as the fact that the verse describes Birkas Kohanim as "ואני אברכם" - "I (God) will bless the Jewish people." He also explicity connects this explanation with the 60 words drasha.