It's the reflexive pronoun. Fermarsi
Reflexives need the pronoun to agree with the subject. If you aren't familiar with them it's worth a read. This was one of those pieces of grammar I wasn't able to intuitively pick up.
yes it is necessary Fermarsi is a different verb to Fermare Fermare - io ho fermato la macchina = I stopped the car Fermarsi - la machhina si è fermata = the car stopped itself (in the sense that it broke)
ps if you wanted to say the watch stopped something you would need the auxiliary Avere - L'orologio ha fermato qualcosa
I'll try minimise the jargon. Use avere when the verb is doing something to an object e.g. kicked a ball. Use essere when the verb isn't doing something to an object e.g. the party started. Use essere instead of avere when the verb is doing something to an object which is itself e.g. I wake up (myself, not someone else)
So did I. This would have been correct 100 years ago but is not now accepted. English has simplified this construction out of existence. It is only found in old songs("Joy to the world, the Lord is come.") and literature, sometimes poetry. I guess we are just out of our time and the world has passed us by.
The normal Past simple would be "the watch stopped." Your sentence is possible as an emphatic structure, known as emphatic do, and usually used to correct someone:
"The watch didn't stop" - "The watch did stop"
But there's no suggestion of that meaning here.
I know Spanish has a special structure equivalent to emphatic do involving sí (yes) - "El reloj sí se paró"
Perhaps Italian has something similar.