赠 is more literary/formal than 送 (sòng, to give). You'll see it in 赠品 (zèngpǐn, complimentary gift), 赠送 (zèngsòng, to present as a gift), 捐赠 (juānzèng, to donate), and the marketing phrase 买一赠一 (buy one get one free).
Left shell radical, the simplified form of 貝. Cowrie shells were the earliest Chinese currency, so 贝 indexes anything about money, value, or trade - 财 wealth, 购 buy, 贩 peddle, 贵 expensive. To 赠 is to hand over something of value as a gift, so the shell anchors the meaning at the very root.
Right side is 曾, originally a pictograph of a tiered steamer, supplying the sound: céng → zèng with a regular voiced/voiceless shift. Same phonetic powers 增 increase, 憎 hate, 僧 monk, 蹭 graze. Faintly semantic too - a gift 'adds' value to the recipient, echoing the 'piled-up steamer' image of 曾.
