Hi, James!
"Ci vediamo" is a RECIPROCAL verb ("vedersi"). The form is the same as with reflexive verbs, i.e. requiring a reflexive pronoun. The subject here is "noi" (" we"); therefore, the reflexive pronoun is "ci". Reciprocal verbs are those verbs that are used to convey the sense of "each other", so, basically, the meaning of "ci vediamo" is "we see each other". "Ci vediamo" is of course also used in Italian as a fixed expression: it's an informal greeting, which translates as "See you soon".
Hope this helps!