Not have seen this Shield meter myself though, I've got no hints as to whether it's to do with the durability of the shield or the stamina of Link himself. Personally, I'd prefer it be the latter, it would make you have to use different strategies to battle the opponents in usual games, and depend on how hard they hit, it would take a chunk out of Link's stamina. Anyone with a basic knowledge of Physics will know what if an object connects with another object with force, that a portion of the force moving the original object will transfer into the object it struck. So applying that to Zelda, you'd get the fact that Link would have to hold up a rather heavy weight whilst defending himself, whilst any force that strikes the shield will then transfer into his shield and then into his arm almost instantly. It would make your arm feel tired and weak rather quickly, and it would make holding up the shield harder depending on the amount of time you hold it up and the force behind any attacks that hit it. Just holding it up could drain a small amount of Link's stamina every second.
So as you can see from that paragraph, once the meter had reached 0, Link would have to stop using his shield for a certain amount of time so that his arm could recovery, forcing the player to go on the offensive.
The idea about the durability of the shield seems more likely though, but I'll find it a pain in the *** to make it halfway into a temple just to find out that my shield has broken.