Theory:

He came to the door of a cottage,
In travelling round the earth,
Where a little woman was making cakes,
And baking them on the hearth;

And being faint with fasting,
For the day was almost done,
He asked her, from her store of cakes,
To give him a single one.
Explanation:
 
While he was travelling around the world, Saint Peter reached a cottage in a remote place.
 
house-2151102_1920.jpg
Saint Peter came to a cottage
 
There he came across a little woman baking cakes on a fire oven. In those days, people didn't have electricity or gas. They used fire to cook and keep themselves warm. Similarly, the oven that they used were quite different from the oven that we see these days. People had used fire ovens back then. It is also called a brick oven or masonry oven.

oven.jpg
A traditional fire oven

So coming back to the poem, Saint Peter reached the house of the woman. Unfortunately, he was very weak and tired and felt faint because he hadn't eaten anything. He walked to the door of the cottage and asked the woman to give him a single cake from the plenty she had baked.
 
woman-157177_1280.png
A woman was making cakes
 
Meanings of difficult words:
 
S.No
Words
Meanings
1
CottageA small house, typically one in the country
2
HearthThe part of a brick oven on which the fire rests
3
FaintTo feel weak, as if you are about to become unconscious
4
FastingThe act of abstaining from all or some kinds of food or drink, especially as a religious observance, or at times for medical or experimental reasons
5
StoreAn amount of something that is being kept for future use
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.