。
Would you like to go out for dinner?
Literal
Dinner [to-に] go-out-[negative-polite-ませんか]?
~ませんか as an invitation is the negative-question form literally asking 'won't you do X?' but functioning as 'would you like to?' This is the standard polite invitation pattern — softer and more inviting than the volitional ~ましょう. 出かける ('to go out, set out') is again chosen over plain 行く to emphasize departing for the purpose.