The literal meaning of 借りてきた猫 is a “borrowed cat”, and it refers to a person who is usually quite active but has recently become quiet and subdued, mostly because he/she is intimidated by being in a different environment or situation than they are used to. You have probably felt like this if you ever attended a party where you do not know many people, or if you were ever the new kid in school. But what does this have to do with “borrowed cats” exactly?
Until the beginning of Edo period, cats were rare pets - owning one was a luxury. And cats are good at catching mice, so cats were viewed as valuable animals. Because not everyone could afford a cat, some people would borrow cats to get rid of vermin from their houses.
However, people often found that once their brought the borrowed cat to their house, the cats just froze and refused to move. And, if you have ever owned a cat, you probably know this is because cats hate it when their environments change!
借りてきた猫 has connotation of naturally becoming quiet because one is nervous or feels uneasy. You can use it in both negative and positive ways depending on your sentence - but it’s probably not a good word to use to people of a higher social status or that you don’t know very well. They could take it the wrong way!
It seems like our child started acting very quiet and shy after she changed schools.
Romy seems to act so quiet and shy in front of the guy she likes.