1. What does it mean?
환승연애 uses the image of changing trains to describe switching partners. As an accusation it can suggest a new relationship was lined up before the previous one was clearly ended.
Also written or said as hwanseungyeonae
2. The nuance
A quick new relationship does not by itself prove cheating. The phrase is also the title of a dating reality show, so distinguish the program from a claim about real people's conduct.
3. Examples
“드라마 속 인물은 이별을 정리하기도 전에 환승연애를 시작했어.”
The character in the drama started a new relationship before properly ending the old one.
“헤어진 뒤 빨리 연애한다고 모두 환승연애라고 할 수는 없지.”
Not every relationship that starts soon after a breakup means someone had already lined up a replacement.
Sources
Sinjoeo Korean Slang Dictionary
References support the meaning and usage. Example sentences are written for NoCapWiki.