In Matthew 10:13, Jesus is sending out his disciples to preach the good news, telling them, "If the household is worthy, let your peace come on it, but if it isn't worthy, let your peace return to you." (Matthew 10:13 WEB)
This entire phrase is an idiom in Hebrew, in which the meaning of the words is not the same as the literal meaning. Idioms are culture specific, varying depending on the language and even location. Any phrase that has an understood meaning that differs from the literal meaning is an idiom.
Examples of Idioms include:
Let the cat out of the bag. (USA) Meaning: Tell the secret.
Spill the beans. (USA) Meaning: Tell the secret.
Across the pond. (UK) Meaning: The other side of the Atlantic Ocean.
All that, and a bag of chips. (USA) Meaning: Person thinks they are special.
Last Coca-Cola in the desert. (Mexico) Meaning: Person thinks they are special.
May I lay my life down for you. (Tribe) Meaning: Hello.
Giving Peace. (Hebrew) Meaning: Friendly Greeting
Idioms are one of the reasons why so many English translations of the Bible exist. A word-for-word translation can sometimes be confusing, which is why there are many translations that slightly paraphrase in order to effectively communicate the meaning.
To someone foreign to any given language, idioms will sound weird or unfamiliar when they are translated word-for-word. The meaning is lost in translation because the meaning was derived from a cultural understanding, rather than the literal words.
Likewise, this phrase spoken by Jesus would have been a familiar phrase meaning essentially, "Greet them, and if they are unwelcoming then leave without regret."
And in fact, while Jesus did not say this directly to Paul, he and Barnabus have a similar response when they experienced hostility in a city in Acts 13. In response, they left without regret in order to go to another city that would be more welcoming. "But they shook off the dust of their feet against them, and came to Iconium." (Acts 13:51 WEB)
Jesus is essentially telling his disciples to not waste time on those who are unwelcoming and who are not ready to receive the good news.
The concept of letting your peace return to you makes sense if we consider the example of a physical gift. The Apostles were to give their blessing to those who received them, and if the host refused the blessing then it would naturally remain with them, just as if they had tried to give an item that remained in their hand, having not been taken.
"He who says he remains in him ought himself also to walk just like he walked." (1 John 2:6 WEB)