The noun 機(き) refers to a ‘chance’ or ‘opportunity’. This grammar pattern marks a specific event or time that serves as a good trigger for an action. The phrase has a somewhat formal feel and is often translated as ‘on the occasion of (A), (B)’. Because this pattern is formed with the particles を and に, it emphasizes that the speaker is deliberately using the event (A) as a catalyst for action (B). This often carries the nuance of ‘using (A) as a chance to do (B)’ or ‘seizing the opportunity presented by (A) to do (B)’.

を機(き)に is placed directly after the noun that represents the opportunity (A). The clause that follows describes the action (B) that was taken as a result of this opportunity.

• 昇進(しょうしん)を機(き)に、広(ひろ)いアパートに引(ひ)っ越(こ)すことにした。
  I decided to move to a more spacious apartment, using my promotion as the opportunity.
• パソコンが故障(こしょう)したのを機(き)に、欲(ほ)しかった最新(さいしん)モデルを買(か)った。
  I used my computer breaking down as a chance to buy the latest model I'd been wanting. (It was the perfect excuse to finally upgrade)
• 大学(だいがく)卒業(そつぎょう)を機(き)に、一人旅(ひとりたび)をしてみようと思(おも)う。
  I think I'll take the opportunity of my university graduation to try traveling by myself.
• 来月(らいげつ)、新(あたら)しい町(まち)に引(ひ)っ越(こ)します。これを機(き)に、古(ふる)い家具(かぐ)は全部(ぜんぶ)捨(す)てるつもりです。
  Next month, I'm moving to a new town. I plan on taking this opportunity to throw away all my old furniture.

This grammar pattern is commonly used when the action in (B) is a conscious decision made by the speaker, for which the event in (A) serves as the primary motivation or a convenient pretext.