Filthy, "filthy," is another word for dirty.  “肮脏的”是“脏的”另一种表示。

So, the rug on the floor looks dirty, so she gets "out the vacuum" and vacuums it. 地板上的地毯脏兮兮的，所以她拿出真空吸尘器清理了一下。

The vacuum, "vacuum," is a machine that removes dirt from the floor – from carpet or rugs -  by sucking the dirt out.   真空吸尘器，是通过吸附，来清除地板或地毯污垢的机器。

Vacuum can also be a verb, so when you say, "I'm going to vacuum my rugs,"  “真空”也可以作动词，当你说“我要把地毯真空一下”，

you mean you're going to use a vacuum to clean your rugs. 意思就是，你要用真空吸尘器来清洁你的地毯。

Lucy picks "up around the room and puts things away."   露西在收拾房间，把东西都收起来放好。

To pick up can mean to take with your hand and grab something and lift it,  To pick up意思就是用手抓住一些东西并拎起来，

but in this case, to pick up means that Lucy is going around and putting things where they  belong  但是，这种情况，意思就是露西四处走动，把东西放回它们所属的地方

–taking the newspaper out and throwing in away or making sure that the table looks nice, that it's been dusted.   ——报纸扔掉，或者确保桌子看起来很整洁，一尘不染。

This is to pick up - to take things that don't belong in the room and move them somewhere else.  这就是收拾——拿走不属于房间的东西，放到别的地方。

When you move them somewhere else, we say that you are putting them away.  当你把它们放到别的地方，我们就说你正在把它们收起来。