This is the second verse of an old rhyme set to a dance tune. The first verse is:
Half a pound of tuppeny rice,
Half a pound of treacle,
That's the way the money goes,
Pop! goes the weasel.
People have different theories as to what all of this means, but it's most likely to be about how easily and quickly your money can disappear. The line "Pop! goes the weasel" could have any number of meanings: it could be a purse made of the skin of a weasel which had a snap fastening that went "pop!" when you opened or closed it; or it could be Cockney rhyming slang for "coat" -- "weasel and stoat" -- and to "pop" something means to pawn it, to take it to a pawnbrokers in exchange for money in the hope of being able to buy it back later.
The "Eagle" was a famous public house and dance hall, a place where you could easily spend a lot of money -- that much is certain. City Road (capital letters) is an actual road in London. It did have a pawnbroker's at one end (that would support the pawning the coat theory), but I'm pretty sure it had other shops as well.
There are many other verses to this rhyme, although there is no definitive version. Some seem to be about everyday things people of the time often needed, others seem to be about drinking (or spending money on drink). It seems that 150 years ago, families had the same kinds of problem we have today...