The Gemarra asks why the bracha of "Seem Shalom" is after Bircas Cohanim. I am not clear what the reason is given in the Gemarra. The Kohamim will put Hashm's name on Klal Yisrael and Hashm will bless them. Is Hashm blessing Klal Yisrael with the final bracha of "Seem Shalom"? Or is Hashm blessing them through the brachas of Bircas Kohanim? If Hashm is blessing Klal Yisrael through the Bircas Kohanim then what does that have to do with the last bracha of Seem Shalom?
Avraham Sacks, Ramat Beit Shemesh
The answer is that Hash-m finishes Birkas Kohanim with "v'Yasem Lecha Shalom," and the last Berachah instituted by the Anshei Knesses ha'Gedolah is regarding Shalom. This gives the effect, as the Maharsha says, "ha'Kol Holech Achar ha'Chasimah" - "Everything is based on the ending (of the Berachah)," which is that Bnei Yisral should have Shalom. Just as Hash-m said that the last Berachah of Birkas Kohanim should finish with Shalom, so too the Anshei Knesses ha'Gedolah realized that it is best to finish Shemonah Esreh with the blessing of Shalom.
All the best,
Yaakov Montrose