Он является одним из главных исполнителей российской сцены и настоящим народным артистом, песни которого разлетаются на цитаты.
He is one of the most well-known Russian entertainers and a true people's artist, whose songs fly away in quotations.
The meaning is clear: people adopt phrases and excerpts from his songs as "household phrases". It's as if phrases from his songs are turning into birds and flying away on their own, they are so strong.
The only English alternative to Russian "fly away in quotations" that I was able to recall is "his songs give rise to household phrases".
I wonder if there are more English expressions to carry across the same meaning.