Let's start from the beginning: in Webber's short film we can clearly see, during Webber's childhood, his father:
It doesn't take a genius to grasp the obvious similarity to Woodie:
Both have caveman foreheads, orange hair and beards, and wear suspenders.
Now, some might argue: But webber's father seems so smart!! Woodie is stupid! Wait and listen...
Back to Webber's short film: as you can clearly see, Webber has become the monster he is after experiencing the "side effects" of Wagstaff's demonic machine:
Webber's father (Woodie) would have become a Werebeaver NOT for a generic curse ... but because of Wagstaff's invention too!
What an ironic fate, Woodie became a chimera like his son in the end!
And Lucy? Lucy would be none other than the spirit of Woodie's deceased wife. Which wife? THIS wife -->
At this point, Woodie, since he was a man of great intelligence, went mad: he had lost his wife and child, and had been turned into a monster.
He cut his head, became a fool and an idiot and found in the hermitism of the Canadian forests the only salvation from madness.
It would also explain Woodie's hatred of birds: Webber probably wanted to play with his father to catch them (as well as butterflies and insects) but Woodie, when he still could, never played with him. And the vague memory of it makes Woodie hate anything that flies.
This is the story of Woodie: a man destroyed and mentally annihilated, who sought the only opening in dementia to not go completely crazy in the face of the horrors of life. Congratulations Klei, for having brought Don't Starve back to the horror and nihilistic dimension of its origins.
Edit: Licet_insanire pointed out two things to me:
1) The fact that Woodie doesn't have glasses makes perfect sense: he lost them during one of the first transformations; and the stupid expression is because he is nearsighted.
2) Lucy, when alive, had red hair. And Lucy's predominant color is red. I don't think it's a coincidence at all.