"Shall I cut the fruit?"
切りませんか implies that you're asking permission, whereas if you say 切りましょうか it's as if you've already decided you're going to cut the fruit, it's just a matter of when. "Shall I do it now?"
the tiles in DL Japanese are really frustrating. it's always partial words. never, for example, would you see tiles like "merci" "beau" "coup" like you do here with 「きり」 「ましょ」 「う」
「切りましょうか」 has an ultimately different meaning than "shall I cut the fruit". It sounds like "Shall we cut the fruit".
Though my primary instinct was to suggest 「くだものを切りますか？」, I do not know which of it and 「くだものを切りませんか」 is correct, though I instinctively would translate the latter as "Are [you/we] not gonna cut the fruit?"
Still. If we talk about nuances this sentence is too vague. "Shall I cut the fruit" may also try to imply that among a group, am I the one who will cut the fruit. In that case I could have also said 「私がその果物を切りますか」.