。
She lives next door to him.
Literal
She [topic-は] his [genitive-の] next [at-に] is-living.
住む ('to live, reside') is one of the verbs that takes its location with に rather than で — に marks the static residence (where one is settled), not で (where an action takes place). The ~ている form 住んでいる reads as the ongoing/resulting state of having taken up residence: 'is in the state of living.' The plain 住む (without ~ている) would sound oddly inchoative — 'will live, take up residence.'