Theory:

Once, when the good Saint Peter
Lived in the world below,
And walked about it, preaching,
Just as he did, you know,
Explanation:
 
The fourth stanza introduces one of the two principal characters of the ballad: Saint Peter. Saint Peter was one of the Twelve Apostles of Jesus Christ and one of the first leaders of the early Church. Peter is also known as Simon Peter, Simeon, Simon, or Cephas.

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Jesus with his 12 apostles

So, according to the speaker, the story takes place when the good Saint Peter had lived on earth. The speaker uses the word "once" to stress the idea that the legend has no factual data.

So, coming back to the poem, when Saint Peter lived on earth, he took it upon himself to travel around the world and preach to people.
 
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The legend revolves around Saint Peter
 
Meanings of difficult words:
 
S.No
Words
Meanings
1
Saint PeterOne of the Twelve Apostles of Jesus Christ
2
PreachingThe act of delivering a sermon (religious address/talk) to an assembled group of people, typically in church
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.