olvidar vs olvidarse
Olvidar is to forget something. Olvidarse de is the same thing in a more personal way — and 'se me olvidó' is the structure that makes it accidental.
olvidar
To forget — direct object, you as the subject.
Olvidé su nombre.
I forgot his name.
A simple statement. You are the one who forgot.
Nunca olvidaré aquel día.
I'll never forget that day.
Emotional memory. Also 'olvidar' = to put out of mind.
Olvídalo.
Forget it.
Let it go. A common answer to 'never mind'.
olvidarse
To forget — reflexive, with olvidarse de, or the accidental 'se me olvidó'.
Me olvidé de las llaves.
I forgot my keys.
Olvidarse de + thing. You are still the subject.
Se me olvidó llamarte.
I forgot to call you.
'Se me olvidó' puts the forgotten thing as the subject. It happened to me, it wasn't my fault.
Se nos olvidaron los billetes.
We forgot the tickets.
The verb agrees with the forgotten item (billetes → olvidaron).
Tricky bit
'Olvidé las llaves' (I left the keys behind) can mean physically leaving them somewhere. 'Se me olvidaron las llaves' is the softer, no-blame option.