so she asks her "supervisor if she wanted to" go with her to lunch.  所以她问她“上司是否愿意”和她一起去吃午饭。

To pack, "pack," a lunch means to make food at home,  打包，“打包”午餐意味着在家里做食物，

put it in a bag or a small box, and bring it to work; that's packing a lunch. 把它放在一个袋子或一个小盒子里，然后带着它去工作；那就是打包午餐。

Lucy says that she asked her supervisor if she wanted to have lunch together. 露西说，她问她的上司是否想一起吃午饭。

A supervisor, "supervisor," is another name for your boss - the person who tells you what to do. “上司”是你老板的别称——告诉你该做什么的人。

In my life, that would be my wife! 在我的生命中，那是我的妻子！

Lucy says that she and her supervisor "head down the street to a sit-down restaurant."  露西说，她和她的上司“沿着街道走向一家可以坐下吃饭的餐馆。”

To head, "head," somewhere means to go somewhere or to go in the direction of something.  去往，“去往”某处意味着去某个地方或朝某个方向前进。

To "head down the street" just means to go down or to walk,  “沿着这条街走下去”，就是说一直开或一直走，

in this case, on "the street to a sit-down restaurant,"  在这种情况下，“走在去往可以坐下吃饭的餐馆的街上，”

a restaurant where you can have a more formal meal. 一家可以提供更正式膳食的餐馆。

"The hostess shows" them "to a table in the back of the restaurant." “女领位”把他们“带到位于餐厅后面的桌位上。”