expression, noun
expression of gratitude; way of thanking
A set phrase meaning 'a way of expressing thanks' or 'words of gratitude'. Often used when one feels that a simple 'thank you' is insufficient.
こんなに親切にしていただいて、お礼の申し様もありません。
You have been so kind; I have no words to express my gratitude.
お礼の申し様もないほど感謝しています。
I am so grateful that I cannot find the words to thank you.
お礼 is a simple 'thank you' or token of gratitude, while お礼の申し様 refers to the manner or words of expressing thanks, often used in formal apologies for insufficient thanks.
感謝の言葉 is a more general term for 'words of gratitude', while お礼の申し様 is a fixed, somewhat formal expression often used in set phrases like お礼の申し様もありません.
A formal phrase combining お礼 (thanks) with 申し様 (way of saying). The exact historical derivation is uncertain, but it is a conventional polite expression.