The 'shelter' radical wraps the top-left - a side-view of a roof leaning against a wall. It indexes 床 in the building/place family alongside 店 shop, 库 warehouse, 庭 courtyard, 庄 village. It signals 'something kept indoors under a roof.'
Inside sits 木 'wood' - a pictograph of a tree, here naming the material the bed is made from. Shelter + wood = the wooden frame you sleep on indoors. The compound is purely semantic, no phonetic. Same shelter-over-wood logic appears in many simple nouns; here it gives the directly readable 'wooden-thing-under-a-roof' = bed.
