The grammar is “A に B はいらない” meaning “B is not necessary for A”.
The A is 別れ meaning “farewells” and the B is 多くの言葉 meaning “many words” (多くの followed by a noun means “many”. In daily conversation たくさんの is commonly used as 多くの sounds formal and non-colloquial in this usage. 言葉 means “words”).
The sentence ending particle さ is usually used by optimistic or snobbish men to make the speech sound cool, dry and frank. The most famous anime character who often uses this is Kazuhiko Hanawa from anime series “Chibi Maruko Chan”.
So 別れに多くの言葉はいらないさ means “You don’t need many words for farewells” in a cool way.