馆 splits cleanly down the middle: 馆 = 饣 + 官. On the left stands 饣, the running-hand abbreviation of 食 shí, the food radical, three quick strokes that once drew a covered vessel of grain. On the right sits 官 guān, an official, a public office, which here does double duty: it carries the sound guān → guǎn and lends the sense of something public, run for others rather than for the household. This is a textbook pictophonetic, the food side telling you what kind of place, the 官 side telling you both how to say it and that the place is a公-facing one.

Read the two halves together and the original picture surfaces: a place where the state feeds its guests. The earliest 馆 were post-houses and official hostels along the imperial roads, where envoys, couriers, and travelling officials were lodged and fed at public expense. Food plus office equals the relay station. The Shuōwén glosses 馆 as the lodging where guests are received, and the food radical is no accident, because hospitality in that world meant a hot meal as much as a roof.

From that relay-house root the meaning spread outward by analogy. Any building set up to serve the public could take the suffix 馆: feed people and it is a 饭馆, shelve books and it is a 图书馆, display relics and it is a 博物馆, house honored guests and it is a 宾馆. The food radical faded from the foreground, but the logic held: 馆 is always a roofed place run for visitors.

Japanese borrowed the character as kan (館), keeping the very same building sense, which is why 図書館, 博物館, and 大使館 read across the two languages with only the script standing between them.

The food radical earns its place here. Before 馆 named libraries or embassies, it named the spot you went to eat, and that street-level sense is still the first one a beginner meets. 饭馆 is the everyday eatery, smaller and more casual than the grand 饭店.

Here 馆 circles back to its origin. The relay-house that fed travelling officials becomes the modern 宾馆 (a hotel for guests) and the 大使馆 (the embassy, literally the great-envoy's house). The 官 inside the character, an official, an arm of the state, surfaces again in these diplomatic and hospitality senses.

馆 is a workhorse noun rather than an idiom character, so set phrases built on it are few, and they cluster around lodging. 离宫别馆 and its inversion 别馆离宫 name the ruler's lodges away from the palace; 茶馆酒肆 names the public houses of the common town. Each frames 馆 the same way: a roofed place where people are received, fed, and put up.

Picture a dusty imperial road at dusk. A courier reins in his horse at a low building with a lantern over the door. Inside, an 官, an official in dark robes, ladles hot rice from a covered pot into the traveller's bowl. That single scene holds both halves of the glyph: 饣, the steam rising off the food on the left, and 官, the state official on the right who runs the place. 馆 is a building where the public is fed and housed.

Now fast-forward. The lantern stays, the official fades, but the building keeps serving the public, only the contents change. Pour in books and the relay-house becomes a 图书馆. Fill it with relics and it is a 博物馆. Lay out beds for guests and it is a 宾馆; raise a foreign flag and it is a 大使馆. Whenever you meet 馆, ask what is being served inside, and the word in front of it will tell you.