If you're studying something intently, you're probably leaning forward and you're oblivious to whatever is going on around you. Your attention is focused like a laser.
If you're gazing intently into someone's eyes, you're hanging on to every blink of the eye, every nuance of expression. Maybe you're in love. When you do something intently, you do it with intensity. It's not a coincidence that intently sounds a lot like intensity, as they both come from the Latin intendere, made up of in- ("toward") and tendere ("stretch"). So when you're listening intently to a Bartók string quartet, you're stretching and straining to hear every note.