The 実 here emphasizes that the writer isn't just talking about acorns, but more specifically about the meat inside the acorn. If you think about it more, it makes more sense -- the text is describing the origins of the character 白 as a symbolic rendering of the meat of the acorn nut, in reference to the off-white color. The text here can't be talking about the outside of the acorn, since that's not white -- it's either green or brown, sometimes even light tan, but certainly not off-white.
Looking up 実 (mi) in both Shogakukan Kokugo Dai Jiten and Daijirin (scroll down for the み reading), the first and second senses describe a plant's fruit or nut / seed, but the fourth sense also notes:
Since a どんぐり is already a nut, it doesn't make sense to parse どんぐりの実 as "acorn's nut" -- what this phrase means instead is "acorn's content → the inside or meat of an acorn".