Feathers radical on the right, the indexing component. Pictograph of two folded wing-feathers side by side. It places 翅 squarely in the bird-flight family: 翔 soar, 翼 wing, 翱 fly high. Whenever 羽 appears as a component, expect feathers or flight.
Upper-left 支 zhī supplies the sound with a regular shift zhī → chì (an old palatalisation of the initial). It also lends a faint image: 支 is a branch held in the hand, and a wing is a branch-like limb extending from the body.