The song is about working in the mill and supplementing wages with poaching activity,leading to transportation.The weasel is a part of the weaving machine which,every thousand or so spins,gave a popping sound.
Apart from the known verses,the song continues..
My mother taught me how to sew,and how to thread the needle,
Every time I pricked my hand,pop goes the weasel.

You may try and try to spin and never make anything regal,
Roll it up and throw it away,pop goes the weasel.

I went hunting with my dog,I knew it wasn't legal,
The warden caught me with the goods,pop goes the weasel.

I swore I'd never hunt nor sport,but the warden looked at my beagle,
You can tell that to the court,pop goes the weasel.

They put me on a transport ship and fed me on black treacle,
Mouldy rice and weevily bread,pop goes the weasel.

So half a pound of tuppeny rice,half a pound of treacle,
That's feeding for a convict girl,pop goes the weasel.