Because you are using the verb "romperSE", which is reflexive.
romper = to break something (usually deliberately)
romperSE = something breaks
The subject of the sentence is not the person who breaks the object, but the object itself. "Las gafas" is 3rd person plural, so we use the reflexive pronoun "se" and the 3rd person plural suffix of the verb "HAN roto".
As for why "me" is usted in the sentence, Alejandro has explained it perfectly.
So, it would be possible to say:
--> (Yo) he roto mis gafas / (Yo) me he roto las gafas (I broke my glasses).
--> Las gafas se me han roto (word by word, "the glasses broke for me / the glasses broke and I was affected by it"; better translated, "my glasses broke").
When we use the second sentence, we technically don't know WHO broke them because the subject is not explicit... But, we infer it. Everyone will understand that it was me who broke the glasses, but we would rather not say it explicitly because we don't want it to seem our fault!! That's why "he roto mis gafas" doesn't sound natural. It also makes it seem like you broke them on purpose.
Another very similar structure using a different verb:
Se me ha olvidado qué tenía que hacer (I forgot what I had to do!)
(olvidar = to forget something; olvidarse = something is forgotten).
Is it clearer now? :)