The Ahavas Shalom explains that the main way to spread peace is through friends connecting with each other and creating meaningful bonds. This is beloved by Hashem and creates many yeshuos, both physically and spiritually.
This is the meaning of the Mishnah (Avos 1:12) that exhorts us to be from amongst the students of Aharon by loving peace and pursuing peace. The Zohar (Chelek 3, 53B) states that Aharon represents the Divine attributes of chesed and rachamim, which are aroused when friends connect with each other. We are meant to emulate Aharon by bringing the type of chesed that he personified into our relationships with our friends.
Accordingly, when the pasuk says that "Aharon's staff for the house of Levi had blossomed", the word "Aharon" represents the middah of chesed, while the word "Levi" represents connection and achdus, as it indicates "levaya” (accompaniment). This means that the middah of chesed blossoms through achdus and connection between friends.
The day of Shabbos also has this bechina of peace, which is why the Zohar (Chelek 3, 176B) calls it "Shabbos Shalom”. That is why Shabbos is a source of blessing.
