Mouth 口 plus 乞 — a pictograph of rising breath, also 'to beg'. The mouth that asks is the mouth that eats. An older character 喫 did the job for centuries; 吃 won by being four strokes cheaper.
吃了吗？'have you eaten?' served as hello for generations — in a country that remembered famine, asking about food was asking about wellbeing. 吃 metabolizes everything: 吃苦 eat bitterness (endure), 吃亏 eat loss (get shortchanged), 吃醋 eat vinegar (be jealous).
A mouth 口 begging 乞 — feed it: EAT.
吃豆腐 'eating tofu' means taking liberties/flirting — order actual tofu carefully in mixed company.