Spassarsela comes from the Italian reflexive verb, spassarsi, meaning to amuse oneself, or to have a good time. Specifically, spassarsela means to live it up, or to have a great time.
I couldn’t find the etymology of this word, but I did find out that passarsela also exists, and when used in conjugation to ‘bene,’ passarsela bene, means to be well off. When used in conjugation with ‘male,’ as in passarsela male, it means to have a hard time.
The Italian verb passare makes up the two verbs stated above. Among its many meanings, passare also mean to spend time with, or to pass time with someone, or doing something.
So in my etymological genius I conclude that spassarsela (spassarsi) comes from passare, and means to spend time living it up hardcore. Don’t forget, conjugate the verb as all reflexive verbs with a direct object!
Me la spasso a Roma. – I live it up in Rome.