The Bnei Yissachar (as well as R' Avraham Azulai in Ma'aseh Hoshev) explains that each day of the Omer a new level of HaShem's light is revealed, in addition to the existing levels revealed on earlier days. So with each additional day of the Omer there is an additional day of light shining upon us.
In other words, on the 12th day of the Omer, 12 days worth of light are being revealed, hence "Today is 12 days of the Omer."
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