The story
From put + -age, perhaps because the questions posed are often introduced with the expression “I put it to you that …”.
Etymology from Wiktionary, CC BY-SA 4.0.
puttage
Today it means: The putting of questions to a witness during cross-examination in accordance with the Rule in Browne v Dunn, specifically, questions that present a version of events that contradicts the witness's own testimony in order to elicit denials or explanations. Full definition →