Here is the juice I made from wringing out a dust cloth.
Let's narrow down our ideas at the meeting.
You've learned that the kanji 絞 means strangle. 絞る, on the other hand, means to wring out or to squeeze something like a towel, or to narrow down ideas.
Your friend wants to wring out a towel, but before she does so she bows (しぼ) to the towel to show her respect for it.