Translation:No, I do not live in Osaka.
it's an explicit indication that the location of Osaka is a topic of conversation, which in English would commonly be communicated through intonation.
"大阪には住んでいません/I don't live in oSAKA", implying maybe there is more to the story of "in Osaka"—perhaps you live not IN Osaka but NEAR it, or you don't like Osaka and would never move there
に is used here as a postposition (comparable to the English preposition in), while は is a topic marker. The topic marker replaces grammatical particles such as が (subject), を (direct object) and に (ndirect object), but since に in this case is a postposition and not a grammatical particle, it carries semantic meaning and thus can't be replaced. Instead we end up with two particles in a row.
大阪 (おおさか) is Osaka, the place you are talking about, the topic of the sentence and therefore you put it at the beginning.
に and は are particles. に points out the fact that you are talking about the place Osaka, cannot be ommited (correct me if I'm mistaken) and は is there to emphasize. It marks the topic of the sentence and is there to suggest that there might be something else about Osaka than the fact that you don't live there.
すんでいません is the negative form of すんでいます, which is the verb for "to live in".
Hope that helped and don't hesitate to add something if I forgot information or if I'm wrong!
This word is a grammatical construction with two verbs, sometimes referred to as the te-iru form, because the first verb 住む (すむ) "to live, to dwell" is in its te-form and 居る (いる) "to exist, to be" is being used as an auxillary verb to modify it.
Te-iru form can be used with many verbs to indicate continuous states. It is similar to the present progressive tense in English, used to say that you are currently doing an action right now (I am reading a book - 本を読んでいる). But it is also used to express certain on-going conditions which do not use present progressive in English (I am married - 結婚しています).
It is also used to express things like living (or not living) in a certain place. "No, I am not dwelling in Osaka (right now)."