"연이 닿는다면" => 우릭가 다시 만나게 될 운명이라면 / (우리에게) 인연이 있다면 => if we're destined to meet / if our karmas are intertwined / if fate will have us cross paths. 즉, "다시 관게가 맺어진다면"을 개인의 의지와 무관한, 불교 사상이나 운명론적 관점에서 표현한 말입니다. 연(緣) is the Buddhist concept of intertwined destiny, of whether two individual beings will ever come in contact. 닿다 means "touch" or "reach". So 연이 닿는다면 is like "if karma has (us / them) reach each other". It is saying "if we are to meet again" with a strong notion that such an encounter is all determined by fate. There is also 인(因), another word in Buddhism referring to the hidden force governing interactions and destinies. And 인연(因緣) is a more concrete concept of such "coming together" of people's destinies. 인 and 연 are more obscure and not used much except in Buddhism and literature, but 인연 is commonly used. For example, 인연이 있으면 또 만나겠지(If it is in our karma, we will meet again") is a rather common phrase with the same meaning as 연이 닿는다면 또 만나겠지.

This expression generally is used when you have to say good bye but want to meet again someday. "연" is karma / relationship and used with verb "닿다" means approach, connect . I usually use this expression like 인연이 닿는다면 다시 같이 일하고 싶습니다