女 depicts a kneeling figure with crossed arms. As radical it places 嬉 in the social-relations family — characters about people interacting. Here it suggests the playful, unguarded side of human company at ease.
喜 supplies both the sound (xǐ drifted to xī) and the meaning of joy — making 嬉 a rare phonosemantic where both sides agree. The combined reading is people enjoying themselves, hence to play and amuse oneself.
