盯 implies an intense, focused gaze - not a casual look. Often used with 着 (zhe) for continuous staring: 盯着看. 盯紧 (dīngjǐn) means to watch tightly/closely.
Left eye radical - pictograph of an eye stood on end (rotated from horizontal so it fits as a tall left-side component). Anchors 盯 as an eye-action. Family: 看 (look), 眼 (eye), 睛 (eyeball), 睡 (sleep), 瞥 (glance). Where 看 is a casual look, 盯 is a locked stare.
Right 丁 supplies the sound directly - dīng, no shift. Same phonetic in 订 dìng (order), 顶 dǐng (top), 钉 dīng (nail), 叮 dīng (sting). 丁 itself depicts a nail with head on top, shaft below. 'Nailing the eye on something' is a happy mnemonic for 盯's locked stare.
