Let's break it down:
* Me: This is the pronoun "me" (I) in Spanish.
* Voy: This is the present tense of the verb "ir" (to go), which in this context means "I am going."
* Enamorando: This is the present participle of the verb "enamorar" (to fall in love). It implies a process of falling in love, rather than a sudden, completed event.
Therefore, the entire phrase "me voy enamorando" suggests a gradual process of falling in love, with the speaker increasingly feeling attracted to someone.