。。
Because we're neighbors. I'll be dropping by all the time.
Literal
Because neighbor-is-[し]. Frequently will-visit [because-から].
Friendly introduction from a new neighbor. だって at the sentence start means 'because, since' — giving a justifying reason. お隣さん ('next-door neighbor') uses the polite お and さん. し closes the first clause as a listing/reason particle. ちょくちょく is a mimetic adverb meaning 'frequently, often, regularly.' お邪魔する ('to intrude, to visit') is the humble/polite fixed phrase for visiting someone's home — literally 'to do an intrusion.' A warm, neighborly greeting.