Origins
When you engage with an idea, you are β at the etymological level β pledging yourself to it.βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ The word comes from Old French engagier, formed from en- ('in') and gage ('pledge, security'). To engage was originally to offer something as collateral.
The gage itself has Germanic origins. It descends from Frankish *waddi, from Proto-Germanic *wadjΔ , meaning 'pledge'. This root branched widely through both Germanic and Romance languages. In English it produced wage (a pledge of payment for work), wager (a pledge staked on an outcome), and wed (to pledge oneself in marriage).
The financial sense came first. In medieval law, to engage land was to pawn it β to put it up as security against a debt. Mortgage belongs to the same world: Old French mort gage means 'death pledge', because the agreement dies when the debt is settled.
Latin Roots
The romantic sense followed naturally. An engagement is a pledge between two people. The military meaning β to engage the enemy β arrived in the 17th century, carrying the idea of committing forces irrevocably, as one commits a pawn.
Modern usage has drifted far from pawning. 'Employee engagement' and 'engaging content' treat the word as synonymous with interest and attention. But the pledge remains underneath.