Theory:

Now, you shall build as the birds do,
And shall get your scanty food
By boring, and boring, and boring,
All day in the hard, dry wood.”

Then up she went through the chimney,
Never speaking a word,
And out of the top flew a woodpecker,
For she was changed to a bird.
Explanation:
 
Since the woman was too selfish to lead a human life, Saint peter turned her into a bird instead. While we often associate birds with flight and freedom, a bird’s life is not as easy as it sounds. For instance, they don't have a permanent shelter. Instead, they will have to build a nest every year. As far as food is concerned, it is difficult to find food, and more often, they do not get sufficient food to eat.

Hence, because of her selfishness and the resulted curse, the woman would have to lead the life of a bird, where she would live in the wild, build nests every season, and would never find sufficient food to eat.

The line "by boring, and boring, and boring, all day in the hard, dry wood" reveals further about the kind of bird she would be turned into. The word "boring" here refers to the act of drilling holes. Hence, the woman would be a bird that drums and drills on hard and dry tree trunks. Yes, a woodpecker!
 
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She would live the life of a bird

Moreover, the repetition used in the phrase "by boring, and boring, and boring" suggests the monotonous and hard life the woman has in store.

Soon after the Saint spoke the words, the woman went up through the oven's chimney and came out through its outlet as a woodpecker. The speaker describes how the woman was changed into a bird in an instant. She never spoke a word because she didn't see the change coming. It was all too sudden and unexpected.
 
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The woman was turned into a woodpecker
 
Meanings of difficult words:
 
S.No
Words
Meanings
1
ScantySmall or insufficient in quantity or amount
2
BoringMaking a deep hole in something hard
3
WoodAn area of land, smaller than a forest, that is covered with growing trees
4
ChimneyA pipe through which smoke goes up into the air, usually through the roof of a building
5
WoodpeckerA bird that uses its strong beak to make holes in tree trunks in order to find insects to eat
Reference:
National Council of Educational Research and Training (2006). Beehive. A Legend of the Northland - Phoebe Cary (pp. 65 - 67). Published at the Publication Division by the Secretary, National Council of Educational Research and Training, Sri Aurobindo Marg, New Delhi.