The literal meaning of 猫を被る is “to wear a cat” , but it actually refers to a person that tries to hide their real personality and pretend they are a quiet person so that the one’s impression is better”.
猫を被る is not actually the original phrase, it evolved that way over time. Originally, the phrase 寝ござを被る, where 寝ござ referred to a “straw mat,” like the kind used for sleeping. However, the sound of 寝ござ and 猫 were very similar - plus cats by nature fit the meaning of the actual phrase, as they often hide their hunter nature and pretend sometimes just act very relaxed and quiet. Thus, over time the phrase slowly turned into the one we use now, 猫を被る.
猫を被る is quite similar to 借りてきた猫 as they both have a general meaning of “being quiet”. However, 借りてきた猫 has more of a connotation of naturally becoming quiet because of nervousness or fear, whereas 猫を被る has a feeling of being quiet on purpose in order to give a good impression. 猫を被る can also have a secondary meaning of “pretending not to know something.”
猫を被る itself is a verb. It’s also used in a way that just points out the action, so it has neither a negative nor a positive connotation. However, much like 借りてきた猫, it’s a more casual type of word so be careful when using it with people in a higher social status from yourself.
Mayumi acts like she is a quiet, gentle person whenever she is with a cool guy.
No matter how gentle or quiet I behave, no one is attracted to me.