To drastically over-simplify the answer: metallurgy advanced to the point where the soldiers could wear their shield on their body. Breastplates were thick enough (and there were enough layers underneath) to make it almost impossible to score a blow through it. And once the breastplates developed their characteristic bulge, it's extremely difficult to score a direct hit anyway (because your lance or whatever is more likely to glance off than to 'stick,' squandering the energy of the blow).

On a similar note, improved metallurgy now meant that armor on other parts of the body -- rerebraces and vambraces in particular -- could be relied upon to keep your arm from getting lopped off if you try to interrupt some guy's axe swing with your forearm. It's not ideal, of course, but if the choice is between a broken bone and getting clobbered in the head with an axe, well, it's really no choice at all.

Freeing up that second hand also means that you can now dedicate that second hand to weaponry. Shields, for the most part, limit you to one-handed weapons. By getting rid of the shield, you can now stop trying to fight with mace and shield and start fighting with a halberd. And a halberd does a much better job of killing people in armor than a mace does, to say nothing of what it can do to less well-armored opponents.

And since there's a fair bit of overlap between what we think of when we hear "plate armor" and gunpowder, it's worth noting that shields of the time were essentially useless against even the most primitive guns. They were really just bits of wood reinforced with some iron; great for blocking swords, terrible for blocking bullets. This would be why shields no longer returned in force even after battlefields came to be dominated by pike and shot formations like the Spanish tercio.