He lured the man over with the offer of private gain.
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啖 is a literary verb 'to eat', often biting into something solid. It is famous from 苏轼's lychee poem. By extension it means to lure someone with bait or profit — to 'feed' them an incentive — as in 啖以私利 ('entice with private gain').
The mouth radical on the left. Eating is done with the mouth, so 口 gives the meaning — it groups 啖 with eating and speaking words like 吃 (to eat) and 喝 (to drink).
Two fires stacked, standalone 炎 meaning blaze. Here it works only as the sound element; the reading has shifted away from yán to dàn, with no fire meaning carried into 啖.
