The chant is a "repeat after me" or "echo" song. The leader at the campfire chants a line, the boys gathered round are to echo it exactly in reply.
There are many, many versions.
The idea was to make the nonsense lines a little longer each time and to vary the elements in order to make them more tricky to remember, which is great fun but is probably why no one can remember the thing exactly the same way now.
In this case, the nonsense probably goes all the way back at least to Shakespeare's King Lear of 1605 ("Child Roland to the dark tower came, His word was still, Fie, foh, and fum, I smell the blood of a British man") and was immortalized in the English fable Jack the Giant Killer of 1711 ("Fee-fi-fo-fum").
"Fie!" is an old word which you use when you smell something really bad, i.e. a stench, a subject of some fascination for young boys.
By beginning with an allusion to the form of these then well known lines, the expectation was set for something familiar to come, but is immediately turned into a tongue twister to set the audience of young boys off their balance and arrest their attention and engage them. It needs to be remembered that boys were then relatively better educated than they are these days, and that it was fun to play loose as it were with the otherwise more or less serious subjects of their school days.
Add some strategically placed stick-banging and tub-thumping and it gets to be quite the scene around the bonfire!
The form is a playground for improv, but most if not all of the versions people now remember have regularized the opening tongue twister by incorporating an "L" in each line and element, as in Flea! Flea fly! Flea fly flow!
This is how I remember it:
Fie!
Fie fly!
Fie fly foh!
Kooma-lotta kooma-lotta kooma-lotta Feast Day!
Oh, no, no, no not the Veesday!
Eeny meany deci-meany Ooo walla walla meeny, Eeny meany, hexa-meany, Ooo walla walla!
Beat belly oaten boaten Bobo benoten botten, Bobo benoten botten, Bobo benoten botten, shhhhh!
