"I live in Osaka."
This comes from a conjugation of the verb 住む (sumu, to live somewhere) First, it is conjugated to its te-form 住んで, which you use to form different kinds of verb conjugations. Then you add いる, to get the present progressive form 住んで いる (to be living). Turn that into its masu-form and you get the polite 住んで います.
My understanding from previous discussions is that it's used for emphasis in the negative and wouldn't commonly be seen in a positive statement, similar to the use of ( wo ) when saying you will eat something, while using ( ha ) to say you won't. I'm far from an expert, but this is what I've gathered from other people who seemed more informed on the matter.
This site helped me understand when to use で or に. Hope it helps you too.
To add to keppinakki1's comment, you will see the the [て-form verb]います/いる conjugation used mainly for a couple of things: 1) actions in progress, as in わたしは 今 ご飯を たべている (I am eating right now) 2) General continuous states of being or facts. For instance, if you wanted to say that you're married, you'd actually use 結婚しています
It's the second one that's at play here.
I was always confused why the multiple choice questions said 住み while the word used is actually 住んでいます. But coming back to this exercise after having advanced past the third checkpoint, i realised Duo pulled a sneaky on us and taught us a professive form if a verb, before we even know what the heck it is.