long stride or stretching radical (no. 54)
廴 represents the idea of a long stride or stretching movement. As a radical, it appears in characters related to extending, progressing, or moving forward.
廴 is a pictograph of a person taking a long step or stretching, with the modern form retaining the extended leg-like shape. Its exact ancient origin is somewhat uncertain, but the association with movement is consistent.
The shape looks like a person with one leg stretched far forward, taking a long stride. This radical often appears in kanji about extending or progressing.
For イン, imagine an inn with a very long corridor where you must take long strides to reach your room: inn -> イン, and the stretching hallway matches the radical's meaning.