"Se me hizo agua la boca"
My guess is that this means "it makes my mouth water." My question is why is the "se" there and please correct my translation if it's wrong.
Well, you are close, but it actually means "my mouth watered (because of whatever - something yummy I smelled). The reason you need the "se" is because the way the expression is constructed in Spanish, it translates literally as "my mouth turned into water."
"Se me" indicates that this thing, the mouth turning into water, happened to me. Thus:
"Se me cayó el lápiz" - 'The pencil dropped from me', or in natural English: "I dropped the pencil".
"Se te zafó un perno" - "One of your bolts came undone"
"Se le perdió el perro" - His/her dog got lost. And so forth.
You are correct.
The "se" is an impersonal expression. It is much like the "se" in the signs that you see in some stores or shops. Se habla Español aquí. One speaks Spanish here or more commonly Spanish is spoken here.
My mouth is drooling.