Saying "the man" and then following up with "the fellow with the name" emphasies that who he was is less important than what he said; the fact that it's Solomon, writing Ecclesiastes (two names widely recognised in the Western canon) serve to emphasise this. EVEN THOUGH it's written by someone famous in a famous book, what he wrote is STILL more important than who he was. That is part of the meaning here. There's another level to it, as well; the author is well aware that he's out on a limb here and his jovial tone makes it clear that he's not taking himself too seriously.

Stripping it down to just "Solomon, writing under the pen name 'Ecclesiastes', said..." changes the meaning.

I suppose he could have explicitly said "Here are some words written by the famous person Solomon in the famous book Ecclesiastes, but the words are more important than who he was and the book he wrote, and I'm telling you this because it makes it clear how much more important the words are than the origin of them, but don't take me too seriously on this folks!" but that seems clunky.