The se me + is more common in Latin America than in Spain but it’s a more impersonal construction.
Generally, there’s no real difference between olvidar and olvidarse. Really, olvidar is “to forget” and olvidarse is more intense like me olvidé is “I completely forgot”.
Grammatically, olvidar can take a direct object, and olvidarse cannot. That makes olvidar “transitive” and olvidarse “intransitive”… so you need a de with olvidarse.
Olvidé las llaves. = I forgot the keys.
Me olvidé de las llaves. = I forgot all about the keys.
With the se me or se te, se le, se nos etc. that’s a much more impersonal construction where the object is the subject there.
In other words… (me) olvidé corresponds to yo.
However, se me constructions are usually 3rd person singular or plural because it’s something that’s happening to the me or te or whatever.
It makes more sense in translation:
Se me olvidó el libro. = I completely forgot the book.
[Literally, “the book forgot itself to me”… meaning that it’s the book’s fault. You’re saying, “the book went and got itself forgotten by me”, which places the blame squarely on the book and not yourself. So olvidó corresponds to el libro.]
And then plural
Se me olvidaron las llaves. = I completely forgot the keys.
[Literally, “the keys forgot themselves to me”… again it’s the keys fault that they were left behind, not yours. It’s a more… fatalistic and passive way to place blame away from yourself.]
So more literally, se me rompió el coche/carro/auto is like “the car up and died on me”. With romperse as “to break (itself)” the indirect object as me, te, le, les, nos would describe who it’s impacting.
Another good example is morir. Regular morir is “to die”. And morirse implies “dying suddenly” or when it’s unexpected. So assume we’re talking about a guy…
Murió. = He died.
Se murió. = He died.
Se me murió. = “He up and died on me”
The se me murió is indicative of grief and a sense of profound loss, that someone’s death impacted the “me” so much and that it was so sudden.
These are grammatically read a little bit differently. Especially when the verb is different.
Acabó. = It ended.
Acabo de hacerlo. = I just did it. [lit. “I ended of doing it”]
Se acabó. = It ran out.
Se me acabó el tiempo. = Time ran out on me.