The imperial kitchen was where the emperor had his meals.
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膳 is a formal/literary term for a meal. In modern usage it mostly appears in compound words: 膳食 (diet/meals), 用膳 (to have a meal, formal), 御膳 (imperial meals). Everyday speech uses 饭 or 餐.
⺼ is the flesh-meat radical (left-side form of 肉). It anchors 膳 to the food-and-body family: a 膳 is a prepared meal, and the meat radical signals the cooked-flesh end of the dining table. Compare 膏, 脯, 腊 — all meat-radical food words.
善 (shàn) supplies the sound, unchanged. There's a strong semantic echo too — a 膳 traditionally meant the carefully prepared, virtuous meals offered to elders or emperors (御膳, 用膳). Good food, good intent. Same phonetic gives 缮 (mend), 鳝 (eel), 擅 (skilled at).
